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Differentiability



The Definition of Differentiability
Differentiability Implies Continuity
Corners
Cusps
Vertical Tangents
Discontinuities
One-Sided Derivatives
Differentiability on Intervals
Pathological Examples
Summary: A Typology of Differentiability Failure



When the Derivative Exists — and When It Does Not


Differentiability is not automatic. The limit definition requires a specific limit to exist and be finite, and many functions fail this requirement at certain points. A corner produces two competing slopes. A cusp sends the slope to infinity from both sides. A vertical tangent gives an infinite slope. A discontinuity removes the foundation entirely.

Each failure mode has a distinct geometric signature, and recognizing them is essential for correctly applying derivative-based analysis. Differentiability also relates precisely to continuity: every differentiable function is continuous, but continuous functions need not be differentiable. The gap between these two properties is wider than it first appears.

Key Terms

Differentiability— the limit defining the derivative exists and is finite
Derivative— the object whose existence differentiability governs
Continuity— differentiability implies continuity, but not the reverse
Tangent Line— exists at differentiable points; absent at corners, cusps, vertical tangents

See All Calculus Definitions →


Differentiability Implies Continuity


If ff is differentiable at aa, then ff is continuous at aa. The proof is direct.

Differentiability Implies Continuity
f differentiable at a  ⟹  f continuous at af \text{ differentiable at } a \implies f \text{ continuous at } a
Learn more about this formula: Differentiability Implies Continuity →


Rewrite f(x)−f(a)f(x) - f(a) as

f(x)−f(a)=f(x)−f(a)x−a⋅(x−a)f(x) - f(a) = \frac{f(x) - f(a)}{x - a} \cdot (x - a)


As x→ax \to a, the first factor approaches f′(a)f'(a) (which exists by assumption) and the second factor approaches 00. Their product approaches f′(a)⋅0=0f'(a) \cdot 0 = 0, so lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a).

This establishes a hierarchy: differentiability is a stronger condition than continuity. Every differentiable function is continuous, but the reverse implication fails. The functions that are continuous but not differentiable at a point form a significant and instructive class.

Corners


A corner occurs when the left-hand and right-hand derivatives both exist as finite numbers but disagree.

lim⁡h→0−f(a+h)−f(a)h≠lim⁡h→0+f(a+h)−f(a)h\lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h} \neq \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}


The graph makes a sharp turn at x=ax = a—two distinct tangent directions meeting at a single point. The function is continuous at aa (no break in the graph), but the two-sided limit of the difference quotient fails to exist because the one-sided limits differ.

The standard example is f(x)=∣x∣f(x) = |x| at x=0x = 0. The left-hand derivative is −1-1 and the right-hand derivative is +1+1. The graph forms a V-shape, and no single straight line can serve as a tangent. More generally, any piecewise linear function with a slope change at a junction has a corner there.

Cusps


A cusp occurs when both one-sided derivatives are infinite but with opposite signs. As h→0+h \to 0^+, the difference quotient approaches +∞+\infty (or −∞-\infty), and as h→0−h \to 0^-, it approaches the opposite.

The graph comes to a sharp point where the curve doubles back on itself with vertical tangent directions from both sides. Unlike a corner, the slopes do not stay finite—they blow up.

The standard example is f(x)=x2/3f(x) = x^{2/3} at x=0x = 0. The derivative f′(x)=23x−1/3f'(x) = \frac{2}{3}x^{-1/3} tends to +∞+\infty from the right and −∞-\infty from the left. The graph has a pointed tip at the origin, sharper than any corner. The function is continuous at x=0x = 0, but the infinite and opposing slopes prevent differentiability.

Vertical Tangents


A vertical tangent occurs when the two-sided limit of the difference quotient exists but equals +∞+\infty or −∞-\infty. Both sides agree on the direction—unlike a cusp—but the slope is infinite.

The standard example is f(x)=x1/3f(x) = x^{1/3} at x=0x = 0. The derivative formula f′(x)=13x−2/3f'(x) = \frac{1}{3}x^{-2/3} gives f′(x)→+∞f'(x) \to +\infty as x→0x \to 0 from either side. The graph passes smoothly through the origin with no sharp point, but the tangent line there is the vertical line x=0x = 0.

A vertical tangent line has undefined slope, so the derivative does not exist in the usual sense. The function is continuous and the graph is smooth, yet the rate of change is unbounded at that point. This is the mildest form of non-differentiability—the geometry is clean, but the numerical value of the derivative is not finite.

Discontinuities


If ff is not continuous at aa, then ff is not differentiable at aa. Since differentiability implies continuity, any discontinuity automatically rules out differentiability.

At a jump discontinuity, the function leaps from one value to another. No tangent line can bridge the gap. At an infinite discontinuity (vertical asymptote), the function blows up—there is no point (a,f(a))(a, f(a)) through which a tangent could pass. At a removable discontinuity, the limit exists but either f(a)f(a) is undefined or differs from the limit; redefining f(a)f(a) to equal the limit restores continuity and may restore differentiability.

Discontinuities are the most straightforward obstruction to differentiability. The other failure modes—corners, cusps, vertical tangents—are subtler because the function is continuous at the problematic point.

One-Sided Derivatives


The left-hand derivative of ff at aa is

f−′(a)=lim⁡h→0−f(a+h)−f(a)hf'_-(a) = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h}


and the right-hand derivative is

f+′(a)=lim⁡h→0+f(a+h)−f(a)hf'_+(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}


One-Sided Derivative
f−′(a)=lim⁡h→0−f(a+h)−f(a)hf+′(a)=lim⁡h→0+f(a+h)−f(a)hf'_-(a) = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h} \qquad f'_+(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}
Learn more about this formula: One-Sided Derivative →


The two-sided derivative f′(a)f'(a) exists if and only if both one-sided derivatives exist, are finite, and are equal: f−′(a)=f+′(a)f'_-(a) = f'_+(a).

One-sided derivatives appear naturally at endpoints of closed intervals, where only one direction of approach is available. They also arise at boundaries of piecewise functions, where different formulas govern the left and right sides. Checking whether the one-sided derivatives match at a boundary is the standard test for differentiability of a piecewise-defined function.
Derivative Notation Defining limit Where it is the natural object
Left-hand f'₋(a) limh→0⁻ [f(a+h) − f(a)] / h right endpoint of an interval; left side of a piecewise boundary
Right-hand f'₊(a) limh→0⁺ [f(a+h) − f(a)] / h left endpoint of an interval; right side of a piecewise boundary
Two-sided f'(a) requires f'₋(a) = f'₊(a), both finite interior points; the standard derivative

Differentiability on Intervals


A function is differentiable on an open interval (a,b)(a, b) if it is differentiable at every point in (a,b)(a, b). No endpoint conditions are needed because open intervals exclude their endpoints.

On a closed interval [a,b][a, b], the convention is: ff is differentiable on (a,b)(a, b) and possesses a right-hand derivative at aa and a left-hand derivative at bb. This mirrors the definition of continuity on a closed interval and ensures that derivative-based theorems—the Mean Value Theorem, Rolle's Theorem—apply on [a,b][a, b].

For piecewise functions defined on adjacent intervals, differentiability at each interior boundary requires two conditions: continuity at the boundary (the pieces connect) and agreement of the one-sided derivatives (the pieces connect smoothly). Continuity alone is not sufficient—f(x)=∣x∣f(x) = |x| is continuous at x=0x = 0 but not differentiable there.
Setting Differentiable means Notes
Open interval (a, b) f differentiable at every point in (a, b) no endpoint conditions — endpoints are excluded
Closed interval [a, b] differentiable on (a, b); right-hand derivative at a; left-hand derivative at b required hypothesis for MVT and Rolle's Theorem
Piecewise boundary at a f continuous at a AND f'₋(a) = f'₊(a) (both finite) continuity alone is not enough — |x| is continuous at 0 but not differentiable

Pathological Examples


The gap between continuity and differentiability extends further than isolated points. The Weierstrass function, constructed in 1872, is continuous at every real number but differentiable at none. Its graph is an infinitely jagged curve with no smooth segments anywhere.

This is not an isolated curiosity. Entire families of continuous-but-nowhere-differentiable functions exist, and they arise naturally in fractal geometry and stochastic processes. Brownian motion paths, for instance, are almost surely continuous everywhere and differentiable nowhere.

These examples demonstrate that differentiability is a genuinely restrictive condition. Continuity guarantees an unbroken graph; differentiability demands a smooth one. Most "natural" functions in calculus—polynomials, trigonometric functions, exponentials—are differentiable on their entire domains, which can create the false impression that differentiability is routine. The pathological cases reveal that smooth behavior is special, not default.

Summary: A Typology of Differentiability Failure


The failure modes covered above—corners, cusps, vertical tangents, and discontinuities—each block differentiability through a different mechanism and leave a different signature on the graph. The table below collects them in parallel, organized by what the one-sided derivatives do, whether the function remains continuous at the problem point, and a canonical example for each. Reading across a row identifies the mode; reading down a column reveals what changes from one mode to the next.
Failure mode Left-hand f'(a) Right-hand f'(a) Continuous at a? Canonical example
Corner finite finite, different from left yes f(x) = |x| at x = 0 (slopes −1 and +1)
Cusp ±∞ ∓∞ (opposite sign) yes f(x) = x2/3 at x = 0
Vertical tangent ±∞ ±∞ (same sign as left) yes f(x) = x1/3 at x = 0
Discontinuity undefined or DNE undefined or DNE no step function at the jump; 1/x at 0