Question about how to define open sets for continuous rational functions I am having difficulty doing the following question using the language of open sets.  
Let $(X,\mathcal{T})$ be a topological space, and let $f,g:X\rightarrow \mathbb{R}$ be continuous functions. 
Let $A=\{x\in X:g(x)=0\}.$ Prove that the function $h:(X-A)\rightarrow \mathbb{R}$ defined by $h(x)=\frac{f(x)}{g(x)}$ is continuous
I know how to show that a rational function is continuous if the functions in both numerator and denominator are both continuous in the sense of how is done in a beginning real analysis course.
For this particular question, for the condition $\{x \in A^{c}:\frac{f(x)}{g(x)}<0\},$ if I want to use an open set to describe the inverse image of the range, how do I account for the two cases where $f(x) > 0$ and $f(x) < 0$ given that $g(x) < 0.$
Thank you in advance.

As restrictions of continuous functions are continuous,
f and g over R - A are continuous.
As g is never 0 over R - A, 1/g is defined and continuous over R - A.
Since the product of two continuous functions is continuous,
f/g is continuous over R - A.
