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.