Are products of 2 continuous maps on topological space still continuous?

Question

i.e. Let $f,g:I \mapsto G$, where $G$ is a topological group. If $f,g$ is continuous, is $f\cdot g$ still continuous? I face this problem when trying to show that $\pi_1(G,e)$ is abelian. Let $f,g$ be two loops; I am trying to show that $f\circ g\sim fg$, but I am confused that whether $fg$ is still a loop.

Answer 1

In the definition of a topological group, there is the fact that the application $G\times G\to G$ given by $(x,y)\mapsto xy$ is continuous. Can you use that fact?