Differentiability at a point (one variable)

A function $f: (a, b) \to \mathbb{R}$ is differentiable at $x_0 \in (a, b)$ if the limit

f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}

exists. This limit, when it exists, is the derivative of $f$ at $x_0$ .

Equivalent formulation

$f$ is differentiable at $x_0$ iff there exists $L \in \mathbb{R}$ such that

f(x_0 + h) = f(x_0) + Lh + o(h) \quad \text{as } h \to 0.

Here $L = f'(x_0)$ and $o(h)$ denotes a function with $o(h)/h \to 0$ .

Implications

Differentiability at $x_0$ implies continuity at $x_0$ .
The converse is false: $f(x) = |x|$ is continuous but not differentiable at $0$ .

One-sided derivatives

Left and right derivatives are

f'_-(x_0) = \lim_{h \to 0^-} \frac{f(x_0 + h) - f(x_0)}{h}, \quad f'_+(x_0) = \lim_{h \to 0^+} \frac{f(x_0 + h) - f(x_0)}{h}.

$f$ is differentiable at $x_0$ iff both exist and are equal.