Let $X$ be a topological space and let $f:X \to \mathbb{R}$ ,$g:X \to \mathbb{R}$ be continuous functions. Show that $fg$ is continuous.

My work:
To show $fg$ is continuous at $x$ for each $x \in X$, let $y=fg(x)$.

To show if $N\_y$ is a neighborhood of $y$, then the pre-image of $y$ is a neighborhood of $x$.

I know that there exists $B\_\epsilon(y) \in N\_y$ so I want to show that $（fg）^{-1}(B\_\epsilon(y)) \in N(x)$

Let $N\_x=（fg）^{-1}(B\_\epsilon(y))$, I want to find an open set in $N\_x$.

Can anyone give me a hint of how to choose such open set or idea how to prove this ?