I’ve run across this problem whilst studying for a comprehensive exam, and would appreciate any hints on a solution to it.

> 
> Let $C:=C\left(\mathbb{R},\mathbb{R}\right)$ denote the space of continuous real-valued functions on the real line. Define a map $\phi:\left(C\times C\right)\rightarrow C$ by $\phi\left(f,g\right):=f\circ g$. Assuming that $C$ is equipped with the topology of pointwise convergence, and $C\times C$ is equipped with the product topology, determine whether or not $\phi$ is a continuous function.
> 
> 
> 

My instinct is to say no, but I don’t have any justification for this. Again, I’d appreciate any suggestions.