Differentiability on an interval

Let $I\subseteq\mathbb{R}$ be an interval and let $f:I\to\mathbb{R}$ (or $\mathbb{C}$). The function $f$ is differentiable on $I$ if $f$ is differentiable at every $a\in I$, i.e. if $f'(a)$ exists for all $a\in I$.

One often distinguishes interior differentiability and endpoint behavior: on a closed interval $[a,b]$, “differentiable on $[a,b]$” may mean differentiable on $(a,b)$ with one-sided derivatives at $a$ and $b$.

Examples:

• $f(x)=\sin x$ is differentiable on $\mathbb{R}$.
• $f(x)=|x|$ is differentiable on $\mathbb{R}\setminus\{0\}$ but not on $\mathbb{R}$.
• $f(x)=\sqrt{x}$ is differentiable on $(0,\infty)$ but not differentiable at $0$ (its right derivative is $+\infty$ in the extended sense).