Let $f$ and $g$ be functions defined on $\mathbb R$. Assume that $f$ is continuous at $0$ and $g$ is discontinuous at $0$. Prove that if $f(0) \neq 0$, then $f*g$ is not continuous at $0$. Does $f(0)=0$ imply that $f*g$ is continuous at 0?
The definition of continuous states that function $f(x)$ is continuous at $x_0$ if and only if for each $\epsilon>0$, $\exists \delta$ such that $x \in dom(f)$ and $|x-x_0|<\delta$ imply that $|f(x)-f(x_0)|<\epsilon$
I'm not sure how to do the first part of the question because this is the first time I have had to use multiplication in the continuous definition. However, for the second part, I believe that it does imply that because it will be continuous getting closer to zero and pull $g$ down to zero as well. But again, I'm not sure how to prove this with multiplication.