Prove that if fk: A in Rn-->Rm be a segment of diff f as on A converging point wise to f:A-->Rm suppose Dfk are continuous and converge uniformly to g. Then f is differentiable and Df=g
That's funny. I have that exact same homework question this week too. (I don't suppose you're in 361 at Penn?) What I did was to look at the partials $\displaystyle \frac{\partial f_{k_i}}{\partial x_j}$ and then, for an arbitrary partial, mimic the proof of this theorem for one variable functions.