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What can you say about the continuity of functions at the point $x_0$?

a) $\varphi(x) = f(x)+ g(x)$

if $f(x)$ is continuous at $x_0$ and $g(x)$ is is discontinuous at $x_0$

b) $\varphi(x) = f(x)g(x)$

if functions $f(x), g(x)$ are discontinuous at $x_0$

I think in a) the function will be discontinuous at $x_0$ and tried to prove it this way:

if $f(x)$ is continuous at $x_0$ then $\exists\lim_{x\to x_0}f(x)=f(x_0)$ and if $g(x)$ is discontinuous at $x_0$ then $\nexists\lim_{x\to x_0}g(x)=g(x_0)$ $\to \nexists lim_{x\to x_0}(f(x)+g(x))=f(x_0)+g(x_0) \to \varphi(x) = f(x)+ g(x)$ is discontinuous at $x_0$

But I have absolutely no idea about b).

I hope for your help!

P.S. Sorry for my bad English.

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    @David Mitra: ok, thank$s$ a lot, I try to think in thi$s$ direction.2011-11-29

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