I was wondering why the statement in the title is true only if the functions we are dealing with are continuous.
Here's the context (perhaps not required): http://i.imgur.com/xbAha.png (The upper equation there is just a limit of two sums, and the lower expression is two limits of those two sums.), and if anyone wonders, that's the original source (a pdf explaining the proof of the product rule).
P.S. In the context it's given that $g$ and $f$ are differentiable, anyway I only provided it to illustrate the question; my actual question is simply general.
A big thanks in advance.