Existence/Boundedness of Partial Derivatives in open set implies continuity

Hi,

If we have $\displaystyle U\subset R^n$, U open, and f:U->R, prove that if all of f's partial derivatives exist and are bounded then f is continuous.

Here is my attempt at a solution:

I have tried approaching this by using the triangle inequality; if the partial exists in the ith direction, then if we restrict the function along that direction then I think it must also be continuous along that direction. From this we can use the triangle inequality, choosing delta_i s.t. |f(x)-f(x')|<E/n for all x' s.t. |x-x'|<delta_i. Doing this over all directions and choosing the smallest delta should let us use the triangle inequality and get continuity.

Is this a valid approach? Also, why is the open set required? To insure existence of the open ball from which to get continuity?