Knowlpedia

Differentiability in one variable

The property of having a finite derivative at a point or on an interval.

A f:I→Rf:I\to\mathbb R is differentiable at a∈Ia\in I if the finite limit

f′(a)=lim⁡x→ax∈I, x≠af(x)−f(a)x−af'(a)=\lim_{\substack{x\to a\\x\in I,\ x\ne a}}\frac{f(x)-f(a)}{x-a}

exists. It is differentiable on II if it is differentiable at every point of II, with one-sided limits used at endpoints of an interval.

Examples
  • Every is differentiable at every real number.
  • The function f(x)=∣x∣f(x)=|x| is not differentiable at x=0x=0.
Remarks

Differentiability is stronger than continuity, as recorded in . It interacts with order and shape through results such as .