4
$\begingroup$

Is it true that if $\frac{d}{dx}f(x)$ is continuous, then $f(x)$ is continuous too?

If not, can you give a counterexample?

$\endgroup$
  • $\begingroup$ Have you tried relating the definition of the derivative to the definition of continuity? $\endgroup$ – Mark Bennet Jun 8 '13 at 14:14
  • $\begingroup$ Do you mean the derivative in the sense of distributions? $\endgroup$ – Siméon Jun 8 '13 at 14:19
14
$\begingroup$

Just the fact that your function $f(x)$ is differentiable is enough to prove that it is continuous. The derivative $\frac{d}{dx}f(x)$, need not even be continuous. Please have a look here http://www-math.mit.edu/~djk/18_01/chapter02/proof04.html

$\endgroup$
5
$\begingroup$

To be differentiable at a point $a$, a function must also be continuous at that point $a$. In your question, this holds for all $a\in \mathbb{R}$.

$\endgroup$

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.