Continuity of $f \cdot g$ and $f/g$ on standard topology. Let $f, g: X \rightarrow \mathbb{R}$ be continuous functions, where ($X, \tau$) is a topological space and $\mathbb{R}$ is given the standard topology.
a)Show that the function $f \cdot g : X \rightarrow \mathbb{R}$,defined by
$(f \cdot g)(x) = f(x)g(x)$
is continuous.
b)Let $h: X \setminus \{x \in X | g(x) = 0\}\rightarrow \mathbb{R}$ be defined by
$h(x) = \frac{f(x)}{g(x)}$
Show that $h$ is continuous.

The central fact is that the operations $$p:\ {\mathbb R}^2\to{\mathbb R},\quad (x,y)\mapsto x\cdot y$$ and similarly $q:\ (x,y)\mapsto {\displaystyle{x\over y}}$ are continuous where defined and that $$h:\ X\to{\mathbb R}^2,\quad x\mapsto\bigl(f(x),g(x)\bigr)$$ is continuous if $f$ and $g$ are continuous.
It follows that $f\cdot g=p\circ h$ is continuous, and similarly for ${\displaystyle{f\over g}}$.
