# The limit of the sum is the sum of the limits

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.

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It's not true that the limit of a sum is the sum of the limits only if the functions are continuous. It is true if the latter limits exist. Such is the case if you're dealing with continuous functions. –  David Mitra Sep 12 '12 at 16:14

Take $$f(x)=\cases{0 & for x<0\\ 1 & for x \ge 0}$$ and $g(x)=1-f(x)$. Note that both $f$ and $g$ are discontinuous at $x=0$. Then $\lim_{x\to 0}(f(x)+g(x))=\lim_{x\to 0}1 = 1$, but neither $\lim_{x\to 0}f(x)$ nor $\lim_{x\to 0}g(x)$ exists, and thus also not their sum.