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.

(a) Given $a \in X$, as $f$ is continuous for all sequencie $(x_k) \subset X$, where
$x_k \to a $ implies $f(x_k) \to f(a)$, like wise
$g(x_k) \to g(a)$. As $(x_k) \subset \mathbb{R}$ we have
$$f(x_k)g(x_k) \to f(a)g(a), \ x_k \to a.$$
Therefore $f.g(x) \to f.g(a), \ x \to a.$
