# What is meant when $f:[a,b] \to \mathbb R$ is said to be differentiable?

Sometimes I see an exercise like this:

Let $f:[a,b] \to \mathbb R$ be differentiable. (A few more givens here.) Show that $f'$ has such-and-such property.

What is usually meant by that? Should $f'$ be defined on $[a,b]$ with one-sided derivatives in $a,b$? Or should one rather assume $(a,b)$ to be the domain of $f'$?

• "Should f′ be defined on [a,b] with one-sided derivatives in a,b?" Yes. – Did Dec 7 '13 at 7:46

Usually this would mean that $f$ has one-sided derivatives at the endpoints.
That's the natural result of defining $f'(x_0)$ as the limit $\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$ when $x\to x_0$ is taken to mean that $x$ approaches $x_0$ within the domain of the function.
You can also say, equivalently, that there is some open interval $I$ containing $[a,b]$ where $f$ admits a differentiable extension $g:I\to\Bbb R$. This is the definition we can transport to higher dimensions.