I think this is probably pretty simple. Is this how its done?
there exits sequence $\displaystyle (x_n)$ that converges to $\displaystyle z_0$. f and g are continuous at $\displaystyle z_0$ so $\displaystyle f(x_n)\rightarrow f(z_0)$ and $\displaystyle g(x_n)\rightarrow g(z_0)$ when $\displaystyle n\rightarrow \infty$. so $\displaystyle \lim_{n\to \infty}(f+g)(x_n)=\lim_{n\to\infty}[f(x_n)+g(x_n)]=f(z_0)+g(z_0)=(f+g)(z_0)$ same with $\displaystyle fg$..