TITLE: Proving that existence of bounded partial derivatives implies continuity of a function.
QUESTION [6 upvotes]: 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.

REPLY [5 votes]: Hint: Apply the mean value theorem coordinate by coordinate to $$f(a+h,b+k)-f(a,b)=\big(f(a+h,b+k)-f(a+h,b)\big)+\big(f(a+h,b)-f(a,b)\big).$$