Differentiability in one variable - Imported Knowlpedia Content

A differentiable function (one variable) is a function $f:I\to\mathbb{R}$ that has a derivative at a point $a\in I$ (meaning $f'(a)$ exists), and it is differentiable on $I$ if it is differentiable at every point of $I$ .

Examples

Every polynomial is differentiable at every real number.
The function $f(x)=|x|$ is not differentiable at $x=0$ .

Remarks

Differentiability is stronger than continuity, as recorded in differentiability implies continuity. It interacts with order and shape through results such as derivative sign implies monotonicity.