Differentiability and uniform convergence

Today I came across a problem in elementary analysis. It is quite interesting, so I thought I'd share it here.

The problem is whether the following statement is true:

If a sequence of differentiable functions converges pointwisely to a function f and if the derivatives converges uniformly to a function g (say all convergence are on some open subset of , for the conclusions are local anyway), then the limit f is differentiable and .

When n = 1 this is a standard theorem in mathematical analysis (see e.g. Rudin's Principle of Mathematical Analysis). However, I have not seen this explicitly stated for higher dimensions in any book, and if one just copies say the proof in Rudin's book line-by-line, one only concludes that the directional derivatives of f exist in every possible direction everywhere. Since in higher dimensions this is not quite sufficient to conclude that f is differentiable, I struggled for a while on whether the result is true as stated. (By the way, the result is trivial if one assumes all are continuously differentiable rather than just differentiable.)

My friends and I finally managed to give a positive answer to the above question, and I thought I'd present the proof here. (The proof is just a simple variant of say the proof in Rudin.)

Suppose is a sequence of differentiable functions in that converges pointwisely to a function f and suppose their derivatives converges uniformly to a function g. Then to argue differentiability at , consider the sequence of functions



It is easy to verify that is a sequence of continuous functions of x, and that they are Cauchy in . But by assumption converges pointwisely to



Since the uniform limit of a sequence of continuous functions is continuous, F is continuous at , so f is differentiable at and . This completes the proof.