1
$\begingroup$

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 $f'$?

$\endgroup$
  • 1
    $\begingroup$ "Should f′ be defined on [a,b] with one-sided derivatives in a,b?" Yes. $\endgroup$ – Did Dec 7 '13 at 7:46
4
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.