If $f,g$ are continuous functions, then $fg$ is continuous? 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 ?

First, define the function $h:X \to \mathbb{R}^2$ by $h(x)=(f(x),g(x))$; since $f$ and $g$ are continuous then $h$ is continuous. 
Now, the multiplication function $\cdot:\mathbb{R}^2 \to \mathbb{R}$, $(a,b) \mapsto a \cdot b$ is continuous. Finally, since $(fg)(x):=(\cdot \circ h):X \to \mathbb{R}$, and the composition of continuous functions is continuous, then $fg$ is continuous. 
