Differentiable Function

A single derivative at one point is useful, but you usually want to differentiate a function across its entire domain. This checkpoint defines what it means for a function to be differentiable on an interval and introduces the derivative as a function in its own right.

The derivative function

Let $f$ be defined on an open interval $I$. If $f$ is differentiable at every point $x \in I$, the rule $x \mapsto f'(x)$ defines a new function, the derivative function of $f$, also written $f'$, $\dfrac{df}{dx}$, or $Df$.

Definition. $f$ is differentiable on $I$ if $f'(x)$ exists for every $x \in I$.

Differentiability on closed intervals

For a closed interval $[a, b]$, differentiability at the endpoints uses one-sided limits: $f$ is differentiable on $[a, b]$ if it is differentiable on $(a, b)$ and both $f'_+(a)$ and $f'_-(b)$ exist.

Higher-order derivatives

If $f'$ is itself differentiable, its derivative $(f')' \eqqcolon f''$ is the second derivative of $f$. Inductively, the $n$-th derivative $f^{(n)}$ is defined whenever $f^{(n-1)}$ is differentiable. In Leibniz notation: $\dfrac{d^n f}{dx^n}$.

A function with $n$ continuous derivatives is called $C^n$. A function with derivatives of all orders is called $C^\infty$ or smooth.

Continuous but not differentiable

Differentiability implies continuity, so every differentiable function is continuous. The converse fails.

Example 1 — corner: $f(x) = |x|$ is continuous everywhere. At $x_0 = 0$ the right derivative is $+1$ and the left is $-1$, so $f'(0)$ does not exist.

Example 2 — cusp: $f(x) = x^{2/3}$ is continuous at $0$, but $\dfrac{f(h)-f(0)}{h} = h^{-1/3} \to \pm\infty$; the derivative does not exist.

Example 3 — differentiable with discontinuous derivative: $f(x) = x^2 \sin(1/x)$ for $x \neq 0$ and $f(0) = 0$ is differentiable everywhere (at $0$ by the squeeze theorem), but $f'$ is discontinuous at $0$.

Summary

• $f$ is differentiable on $I$ if $f'(x)$ exists for every $x \in I$, yielding the derivative function $f'$.
• Closed-interval differentiability uses one-sided limits at the endpoints.
• The $n$-th derivative $f^{(n)}$ is defined iteratively; $C^\infty$ functions have all orders.
• Continuity is necessary but not sufficient for differentiability.