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