6
$\begingroup$

Let $E \subset \mathbb{R^n}$ be open and let $f:E \to \mathbb{R}$. Suppose that ${\partial f \over \partial x_1}, ..., {\partial f \over \partial x_n}$ exist and are bounded in E. Prove that $f$ is continuous in $E$

How in the world do I prove this? I thought that the existence of partial derivatives did not imply continuity of the function. How does the condition that each partial is bounded affect that?

Any hints or possibly a sketch of the proof would be welcome. Thanks.

The subscript on the latter partial is an n by the way. I know it's hard to read.

$\endgroup$
5
$\begingroup$

Hint: Apply the mean value theorem coordinate by coordinate to $$f(a+h,b+k)-f(a,b)=\big(f(a+h,b+k)-f(a+h,b)\big)+\big(f(a+h,b)-f(a,b)\big).$$

$\endgroup$
  • $\begingroup$ (Then generalize to $n$ coordinates) $\endgroup$ – Pedro Tamaroff Feb 8 '14 at 3:51
  • $\begingroup$ Aren't you only proving that $f$ has all directional derivatives? $\endgroup$ – Sha Vuklia Apr 9 '17 at 8:41
  • $\begingroup$ @ShaVuklia: Nothing to do with directional derivatives here. We're trying to prove continuity by showing $\lim\limits_{h,k\to 0} |f(a+h,b+k)-f(a,b)| = 0$. $\endgroup$ – Ted Shifrin Apr 9 '17 at 15:40

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.