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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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For b), consider $f=g$ with $f(x)=\cases{1,&x>0\cr -1,&x\le0}$. –  David Mitra Nov 29 '11 at 14:06
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For a), suppose that $\varphi$ and $f$ are both continuous at $x=x_0$. Then $\varphi-f$ is continuous at $x=x_0$ (contradiction). Therefore, $\varphi$ cannot be continuous at $x=x_0$. –  Bill Cook Nov 29 '11 at 14:57
    
@BillCook: Thank you, your proof is better than mine. –  Dima Nov 29 '11 at 17:22
    
@David Mitra: Can you tell more about the next steps? –  Dima Nov 29 '11 at 17:23
    
Well, $f$ is not continuous at $x=0$. But, $f\cdot f$ is identically 1 and, thus, continuous at 0. So, it's possible for the product of two functions, both of which discontinuous at $x_0$, to be continuous at $x_0$. –  David Mitra Nov 29 '11 at 17:27
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