Continuity of fog and f/g on standard topology.

[vercammen]

PICTURE ATTACHED!

Let \(\displaystyle f: X \rightarrow \mathbb{R}\) be continuous functions, where (\(\displaystyle X, \tau\)) is a topological space and \(\displaystyle \mathbb{R}\) is given the standard topology.


a)Show that the function \(\displaystyle f \cdot g : X \rightarrow \mathbb{R}\),defined by

\(\displaystyle (f \cdot g)(x) = f(x)g(x)\)

is continuous.



b)Let \(\displaystyle h: X \setminus \{x \in X | g(x) = 0\}\rightarrow \mathbb{R}\) be defined by

\(\displaystyle h(x) = \frac{f(x)}{g(x)}\).

Show that \(\displaystyle h\) is continuous.




Please help.

 

[HallsofIvy]

It's impossible to know what help you need if we know nothing about what you can do. Do you know what a "continuous" function is? Do you know, in general, how one can prove that a function is, or is not, continuous? What have you tried and where did you hit a snag?