Continuity and Differentiability

Unbroken vs smooth

Class 11 introduced limits and derivatives as tools: a limit described how a function behaves near a point, and a derivative measured how fast the function changes there. Both rested on an intuitive sense of smoothness, without settling exactly what the words mean. This chapter makes that meaning precise, and in doing so separates two properties that look alike on a graph but are not the same.

A function is continuous at a point when its graph has no break there — the limit of the function as x approaches the point matches the value of the function at that point. A function is differentiable at a point when its graph has a well-defined tangent there — the difference quotient $(f(a+h)-f(a))/h$ settles to a single value as h approaches 0 from either side. Differentiability always forces continuity: a well-defined tangent cannot exist where the graph itself breaks. The reverse implication fails: a graph can be unbroken at a point and still have a sharp corner there, so that no single tangent line is defined; $|x|$ at $x=0$ is the standard case, worked out later in this chapter.

That gap between continuous and differentiable is the chapter’s organising fact. Once it is settled, the chain rule, implicit and logarithmic differentiation, parametric and second-order derivatives, and Rolle’s theorem and the Mean Value Theorem all extend the same differentiation apparatus, built on functions already known to be differentiable.

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Continuity at a point

A function f is continuous at x=a when three quantities all agree: the left-hand limit $\lim_{x \to a^-} f(x)$, the right-hand limit $\lim_{x \to a^+} f(x)$, and the value $f(a)$ itself.

The left-hand limit (LHL) and the right-hand limit (RHL) are the same two-sided limit $\lim_{x \to a} f(x)$, split by the direction from which x approaches a: LHL restricts x to values less than a, RHL restricts x to values greater than a. Both measure how the function’s value behaves near a; neither should be confused with the one-sided derivatives — the left-hand derivative and right-hand derivative — defined later in this chapter, which measure the behaviour of a slope, not of $f(x)$ itself.

Continuity at a requires all three quantities to coincide: $\text{LHL} = \text{RHL} = f(a)$. Failure can happen in three distinct ways. The one-sided limits can disagree with each other, so the two-sided limit does not exist. Either one-sided limit can fail to exist outright. Or the one-sided limits can agree with each other but not with $f(a)$, so the two-sided limit exists yet does not equal the function’s actual value at the point. Each of these three cases is a discontinuity at a.

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Algebra of continuous functions

If f and g are both continuous at x=a, then so are the sum $f+g$, the difference $f-g$, and the product $f \cdot g$, with no further condition on f or g required beyond continuity itself. The quotient $f/g$ is continuous at a as well, but only when $g(a) \neq 0$; continuity of a quotient is never automatic at a point where the denominator vanishes, since the quotient itself is undefined there.

Composition inherits continuity along the same lines: if g is continuous at a and f is continuous at g(a), then the composite function f(g(x)) is continuous at a. The two continuity conditions apply at two different points — g must be continuous at a itself, while f must be continuous at the value g(a) that g produces there.

These rules explain why familiar classes of functions are continuous wherever they are defined. Every polynomial is built from sums and products of the identity function and constants, both continuous everywhere, so every polynomial is continuous everywhere on its domain. Sums, differences, and products of continuous functions stay continuous; quotients stay continuous away from zeros of the denominator; and compositions of continuous functions stay continuous.

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Continuity of a piecewise function

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LHD, RHD and differentiability

The left-hand derivative (LHD) and the right-hand derivative (RHD) at x=a are one-sided limits of the difference quotient that defines a derivative. $\text{LHD} = \lim_{h \to 0^-} (f(a+h)-f(a))/h$ restricts h to negative values approaching 0, and $\text{RHD} = \lim_{h \to 0^+} (f(a+h)-f(a))/h$ restricts h to positive values approaching 0.

f is differentiable at a precisely when both one-sided limits exist and are equal; their common value is then f'(a). If LHD and RHD disagree, or if either fails to exist, the two-sided limit $\lim_{h \to 0} (f(a+h)-f(a))/h$ does not exist, and f'(a) is undefined even where f itself is otherwise well-behaved.

LHD and RHD ask a different question from the LHL and RHL introduced earlier in this chapter. LHL and RHL test whether the function value f(x) settles to a limit as x approaches a from one side; LHD and RHD test whether the slope of the secant line — the difference quotient $(f(a+h)-f(a))/h$ — settles to a limit as h approaches 0 from one side. The two pairs are computed from different expressions and answer different questions, even though both are built as one-sided limits.

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LHD/RHD vs LHL/RHL

LHL, RHL, LHD, and RHD are all introduced as one-sided limits, and the shared machinery invites treating them as interchangeable: compute a one-sided limit near x=a, attach whichever label the question asked for, and move on. The expression under the limit sign is what distinguishes them, and skipping past that expression is exactly where the error happens.

LHL and RHL take the limit of f(x) itself as x approaches a from one side — the quantity inside the limit is the function’s value. LHD and RHD take the limit of the difference quotient $(f(a+h)-f(a))/h$ as h approaches 0 from one side — the quantity inside the limit is a ratio built from two function values and the gap between them, not the function value on its own. These are different expressions with different limits, computed for different purposes: LHL/RHL test continuity, LHD/RHD test differentiability.

The consequence is that a function can pass the LHL/RHL test while failing the LHD/RHD test. $f(x) = |x|$ at $x=0$ does exactly this: $\text{LHL} = \text{RHL} = f(0) = 0$, so f is continuous at 0, yet $\text{LHD} = -1$ and $\text{RHD} = 1$ disagree, so f is not differentiable at 0. Passing one test carries no information about the other, because the two tests are built from different expressions.

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Check continuity first

Differentiability is defined by comparing LHD and RHD, which makes it tempting to compute those two one-sided limits immediately whenever a problem asks about differentiability at a point, without first checking whether the function is even continuous there. That order skips a precondition: differentiability at a requires continuity at a. Computing LHD and RHD before continuity is confirmed can produce numbers that look like a valid comparison but describe nothing.

The reason lies in what the difference quotient assumes. Both $\text{LHD} = \lim_{h \to 0^-} (f(a+h)-f(a))/h$ and $\text{RHD} = \lim_{h \to 0^+} (f(a+h)-f(a))/h$ use f(a) as the fixed reference that nearby values f(a+h) are measured against.

If f is discontinuous at a — if $\lim_{x \to a} f(x)$ fails to exist or does not equal f(a) — then f(a) is not the value the function is actually approaching near a, and comparing f(a+h) to it does not measure the slope of anything the graph is actually doing at a. Differentiability forces continuity, established in the atom that follows; read the other way, a point failing continuity cannot pass differentiability, so computing LHD and RHD there answers a question that does not arise.

The efficient check order follows directly: confirm continuity first, and compute LHD and RHD only once continuity already holds.

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Differentiable implies continuous

If f is differentiable at x=a, then f is continuous at a. This direction always holds, and it can be proved directly from the definition of the derivative.

For $h \neq 0$, write $f(a+h)-f(a) = [(f(a+h)-f(a))/h] \cdot h$ — an identity, true for every nonzero h regardless of whether the limit involved exists. Now let $h \to 0$. The bracketed factor $(f(a+h)-f(a))/h$ tends to f'(a), a finite number, because f is differentiable at a by hypothesis. The second factor h tends to 0. A product of a quantity tending to a finite number and a quantity tending to 0 itself tends to 0, so $\lim_{h \to 0} [f(a+h)-f(a)] = 0$.

That limit is exactly the continuity condition, rearranged: $\lim_{h \to 0} f(a+h) = f(a)$, which is $\lim_{x \to a} f(x) = f(a)$ under the substitution x=a+h. So differentiability at a forces continuity at a.

Differentiability is therefore the stronger of the two requirements: continuity is a necessary precondition for it, proved directly above, and every differentiable function is continuous — though the chapter’s later examples show the converse fails.

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Continuous does not mean smooth

An unbroken graph looks like it should have a well-defined tangent at every point, since there is no gap for the tangent line to fail to cross. That intuition treats continuity as though it already contains differentiability, but the proof in the previous atom runs in only one direction: differentiability forces continuity, and nothing in that argument runs the other way.

$f(x) = |x|$ at $x=0$ makes the gap concrete. The function is continuous at 0: $\text{LHL} = \text{RHL} = f(0) = 0$, so the three-way agreement that continuity requires is satisfied, and the graph has no break at the origin. But the one-sided derivatives disagree. $\text{LHD} = \lim_{h \to 0^-} (|h|-0)/h = -1$, computed from the branch $f(x)=-x$ that applies for $x<0$, while $\text{RHD} = \lim_{h \to 0^+} (|h|-0)/h = 1$, computed from the branch $f(x)=x$ that applies for $x>0$. The graph approaches the origin along two straight pieces of different slope, and a break-free graph places no constraint on whether those two slopes agree at the join.

Continuity fixes only the function’s value near a point; it places no constraint on the direction from which the graph arrives there. Differentiability adds that further constraint, and $|x|$ at $x=0$ shows the constraint can fail even where continuity holds.

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Where |x| fails to be smooth

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The chain rule

For a composite function $y=f(g(x))$, the chain rule states $d y/d x = f'(g(x)) \cdot g'(x)$. The outer function f is differentiated first, evaluated at the inner function’s value g(x) rather than at x directly, and that result is multiplied by the derivative of the inner function, g'(x).

The rule can be written in a second, equivalent form that makes the layering explicit. Let $u=g(x)$, so that $y=f(u)$. Then $d y/d x = (d y/d u) \cdot (d u/d x)$: the derivative of y with respect to the intermediate variable u, multiplied by the derivative of u with respect to x. This layered form is the one used repeatedly through the rest of this chapter — implicit differentiation and logarithmic differentiation both proceed by introducing an intermediate variable and applying the chain rule across it.

The two forms are the same rule under different notation. f'(g(x)) is exactly $d y/d u$ evaluated at u=g(x), and g'(x) is exactly $d u/d x$; writing the rule either way gives the same derivative, and the choice between them is a matter of which is more convenient for the function at hand.

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A two-layer chain rule

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Inverse trig derivatives

The six inverse trigonometric functions are defined on principal branches fixed in Chapter 2, and each has a standard derivative formula on the domain where that branch applies.

For sine and cosine inverses, restricted to $|x|<1$: $d/d x \sin^{-1}(x) = 1/\sqrt{1-x^2}$ and $d/d x \cos^{-1}(x) = -1/\sqrt{1-x^2}$. The two formulas differ only in sign, which follows from the identity $\sin^{-1}(x) + \cos^{-1}(x) = \pi/2$ established in Chapter 2: differentiating a constant sum gives derivatives that are negatives of each other.

For tangent and cotangent inverses, valid for all real x: $d/d x \tan^{-1}(x) = 1/(1+x^2)$ and $d/d x \cot^{-1}(x) = -1/(1+x^2)$. The same sign relationship holds, for the same reason: $\tan^{-1}(x) + \cot^{-1}(x) = \pi/2$ is constant, so the two derivatives are negatives of each other.

For secant and cosecant inverses, restricted to $|x|>1$: $d/d x \sec^{-1}(x) = 1/(\left|x\right| \sqrt{x^2-1})$ and $d/d x \csc^{-1}(x) = -1/(\left|x\right| \sqrt{x^2-1})$, again related by $\sec^{-1}(x) + \csc^{-1}(x) = \pi/2$. Each formula holds only on the domain stated; outside it the corresponding inverse function is not defined, so the derivative does not arise.

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Chain rule on inverse sine

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Derivative of the exponential

The exponential function $e^x$ satisfies a property no other elementary function shares: its derivative equals the function itself. That is, $d/d x e^x = e^x$.

This follows from the definition of $e$ as the unique base for which the limit $\lim_{h\to 0} (e^h - 1)/h$ equals $1$. Starting from the general definition of the derivative, $d/d x a^x = a^x \cdot \lim_{h\to 0} (a^h - 1)/h$, the derivative of $a^x$ reduces to $a^x$ itself only when this limit equals $1$ — which happens precisely when $a = e$.

For a general base $a$, the derivative carries a correction factor: $d/d x a^x = a^x \cdot \ln(a)$. Since $\ln(e) = 1$, this general formula reduces to $d/d x e^x = e^x$ exactly when $a = e$, confirming $e^x$ as the base for which no correction term is needed.

This is why $e^x$ is treated as calculus’s natural exponential function: differentiation leaves it unchanged, while every other exponential base’s derivative is expressed as a multiple of itself scaled by $\ln(a)$.

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Derivative of the natural log

The natural logarithm $\ln x$, defined for $x > 0$ as the inverse function of $e^x$, has derivative $d/d x \ln x = 1/x$.

This formula follows from the inverse function rule together with the fact that $e^x$ is its own derivative. If $y = \ln x$, then $x = e^y$, and differentiating both sides with respect to $x$ gives $1 = e^y \cdot (d y)/(d x)$. Since $e^y = x$, this rearranges to $(d y)/(d x) = 1/x$.

The formula holds only for $x > 0$, since $\ln x$ itself is defined only there. It is this result — a power function’s derivative, $1/x$, arising from a logarithmic one — that makes $\ln x$ the tool behind logarithmic differentiation later in this chapter: taking logarithms before differentiating converts awkward products, quotients, and variable exponents into sums and constant multiples, each of which differentiates through $1/x$ and the chain rule.

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Implicit differentiation

Most functions in this chapter so far have been given explicitly, $y = f(x)$, with $y$ isolated on one side. Not every relation between $x$ and $y$ takes that form. When $y$ cannot be isolated — or isolating it would be needlessly awkward — $y$ is still, implicitly, a function of $x$; the equation $x^2 + y^2 = 25$ is one example, since for most $x$-values in $[-5, 5]$ it determines two corresponding $y$-values, each varying with $x$.

Implicit differentiation exploits that dependency directly: differentiate both sides of the equation with respect to $x$, treating $y$ throughout as a function of $x$. Differentiating $y^2$ with respect to $x$ therefore does not give $2y$; it gives $2y \cdot (d y)/(d x)$, because the derivative of $y^2$ with respect to $y$, which is $2y$, must be multiplied by $(d y)/(d x)$ under the chain rule.

Every term containing $y$ picks up a factor of $(d y)/(d x)$ this way. Once both sides are differentiated, the resulting equation is solved algebraically for $(d y)/(d x)$, producing a formula that is typically expressed in terms of both $x$ and $y$ rather than $x$ alone.

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Slope on a circle

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Logarithmic differentiation

Some functions resist direct differentiation not because a rule is missing, but because applying the available rules produces an unmanageable mess. Two shapes recur often enough to warrant a dedicated method: a variable raised to a variable exponent, such as $y = x^x$, where neither the ordinary power rule (constant exponent) nor the exponential rule (constant base) applies; and a long product or quotient of several factors, where direct product- and quotient-rule differentiation multiplies out into an unwieldy expression.

Logarithmic differentiation handles both by taking $\ln$ of both sides before differentiating. Applying $\ln$ converts a product into a sum, a quotient into a difference, and an exponent into a multiplying constant — each far easier to differentiate than the original form.

After taking $\ln$ of both sides of $y = f(x)$, the left side becomes $\ln y$, and differentiating it with respect to $x$ requires the chain rule, since $y$ is itself a function of $x$: the derivative of $\ln y$ with respect to $x$ is $(1/y) \cdot (d y)/(d x)$. Once both sides are differentiated, multiplying back through by $y$ isolates $(d y)/(d x)$ as an explicit formula.

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Differentiating a variable power

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Parametric derivatives

Some curves are not naturally described as $y = f(x)$ at all. A parametric curve instead gives both coordinates as functions of a third variable $t$: $x = f(t)$ and $y = g(t)$, with $t$ ranging over some interval. As $t$ varies, the point $(x, y)$ traces the curve.

The chain rule still delivers $d y/d x$ for such a curve, without ever solving for $y$ as an explicit function of $x$ — a step that is often impossible or algebraically ugly for parametric curves. Since $y$ depends on $x$ through $t$, the chain rule gives $(d y)/(d t) = (d y)/(d x) \cdot (d x)/(d t)$. Dividing both sides by $(d x)/(d t)$, provided it is nonzero, isolates the derivative sought: $d y/d x = (d y/d t)/(d x/d t)$.

This formula treats the parametric curve’s two coordinate derivatives, both with respect to $t$, as the entire input; no explicit $y$-in-terms-of-$x$ relation is ever needed.

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Slope of a parametric circle

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The second derivative

The derivative $d y/d x$ is itself a function of $x$, and like any function of $x$, it can be differentiated again. The result, the derivative of $d y/d x$ with respect to $x$, written $(d^2 y)/(d x^2)$, is the derivative of a derivative: the second-order derivative of $y$ with respect to $x$.

Several equivalent notations denote the same quantity: $(d^2 y)/(d x^2)$, $f''(x)$, and $y''$ all mean the derivative of $d y/d x$, computed the same way any derivative is computed — the two-step process is nothing beyond differentiating twice in succession.

Geometrically, $d y/d x$ tracks the curve’s slope at each point; the second derivative $(d^2 y)/(d x^2)$ records how that slope itself is changing along the curve, the first step toward describing concavity rather than merely steepness.

Nothing about the process stops at two steps. Differentiating $(d^2 y)/(d x^2)$ once more produces the third derivative, written $(d^3 y)/(d x^3)$, and the pattern continues to any order the problem requires.

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Second derivative with the product rule

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Rolle’s theorem

Rolle’s Theorem states a condition under which a differentiable curve must have a horizontal tangent. If a function f is continuous on the closed interval [a,b], differentiable on the open interval (a,b), and satisfies f(a) = f(b), then there exists at least one point c in (a,b) such that f'(c) = 0.

Geometrically, if a smooth curve begins and ends at the same height, its tangent line must be horizontal at some point between the two ends — the curve cannot rise or fall throughout the whole interval without turning back.

All three hypotheses are necessary. Without continuity on the closed interval, the function could jump before reaching b without ever levelling off. Without differentiability on the open interval, the curve could have a sharp corner exactly where the tangent would otherwise be horizontal, and no derivative exists there to equal zero. Without equal endpoint values, the curve could rise steadily from f(a) to a different f(b) with no horizontal tangent anywhere in between. Dropping any one of the three allows the conclusion to fail.

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The Mean Value Theorem

The Mean Value Theorem generalises Rolle’s theorem by removing the requirement that the function take equal values at the two endpoints. Let f be continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$. Then there exists at least one point c in $(a,b)$ such that $f'(c) = (f(b)-f(a))/(b-a)$.

The right-hand side of this equation is the average rate of change of f over $[a,b]$, the slope of the chord joining the points $(a, f(a))$ and $(b, f(b))$. The theorem states that this average rate of change is attained as an instantaneous rate of change at some interior point c; geometrically, the tangent to the curve at $(c, f(c))$ is parallel to that chord.

Rolle’s theorem is the special case $f(a) = f(b)$. When the endpoint values are equal, the chord is horizontal, its slope is zero, and the conclusion reduces to $f'(c) = 0$.

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Verifying the MVT

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Unbroken, then smooth

Continuity asks whether a function is unbroken at a point: whether its left-hand limit, right-hand limit and value there all agree. Differentiability asks the stronger question of whether the function is smooth there, whether a unique tangent line exists. Differentiability always implies continuity, but continuity never guarantees differentiability; $|x|$ at $x=0$ remains the standing counterexample for the chapter, continuous at the origin yet possessing unequal left-hand and right-hand derivatives there.

Each technique introduced after this distinction, the chain rule, implicit differentiation, logarithmic differentiation, derivatives of parametric functions, the second derivative, answers a computational question rather than a conceptual one. None of them revisits what it means for a function to be smooth; each supplies a faster route to a derivative that the limit definition would also produce, given enough algebra.

Rolle’s theorem and the Mean Value Theorem use that computed derivative differently. Rather than reporting the rate of change at a single point, they convert it into a statement about the function’s behaviour across an entire interval: under Rolle’s equal-endpoint condition, some interior point has a horizontal tangent; more generally, some interior point has a tangent parallel to the chord joining the endpoints. The derivative, computed pointwise throughout the chapter, is here shown to carry information about the function as a whole.

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Practice set — continuity

Exercise 5.1 practises testing continuity at a point using the left-hand limit, right-hand limit and the function’s value there, including piecewise functions and functions with a removable gap. It also requires finding unknown constants that make a piecewise function continuous at the point where its two pieces join.

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Practice set — differentiability

Exercise 5.2 practises computing the left-hand derivative and right-hand derivative directly from the difference-quotient limit, then deciding whether a function is differentiable at a given point. The exercise repeatedly returns to corner points of the $|x|$ type, where continuity holds but the left-hand and right-hand derivatives disagree.

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Practice set — chain rule

Exercise 5.3 practises the chain rule on composite functions built from several layers, combining trigonometric, polynomial and nested root functions in a single expression. Several problems stack two or three layers deep, mirroring the worked example on $\sin(\sqrt{x})$, and require differentiating from the outermost function inward.

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Practice set — inverse trig

Exercise 5.4 practises the six inverse-trigonometric derivative formulas, applied both directly and through the chain rule on composite arguments such as $\sin^{-1}(2x)$. Domain restrictions such as $|x| < 1$ matter for full marks, not the derivative formula alone.

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Practice set — exp and log

Exercise 5.5 practises the derivatives of $e^x$ and $\ln x$, combined with the chain rule for composites such as $e^{\sin x}$ and $\ln(x^2+1)$. It builds the fluency in handling exponential and logarithmic composites that the next exercise, logarithmic differentiation, depends on.

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Practice set — log differentiation

Exercise 5.6 practises logarithmic differentiation on functions such as $x^x$, where the variable appears in both the base and the exponent, and on long products or quotients of several power terms. Recognising when to take logarithms first, rather than differentiating the expression directly, is the exercise’s central skill.

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Practice set — parametric

Exercise 5.7 practises finding $d y/d x$ for curves given parametrically, using $d y/d x = (d y/d t)/(d x/d t)$. Trigonometrically parametrised curves, such as $x = a \cos t, y = a \sin t$, appear repeatedly, echoing the worked example.

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Practice set — second derivative

Exercise 5.8 practises finding $d^2 y/d x^2$ for explicit, implicit and parametric functions, differentiating twice in each case. Some problems require verifying that a given function satisfies a stated second-order differential relation.

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Practice set — miscellaneous

The Miscellaneous Exercise combines every technique from the chapter and adds Rolle’s theorem and the Mean Value Theorem. For these problems, the hypotheses are verified on the given interval, then $f'(c) = 0$ or $f'(c) = (f(b)-f(a))/(b-a)$ is solved for the actual value of c.