Differentiability in one variable

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 $I$.

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 .

Examples:

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