Q:

Continuous function as product of two function

I am confronted with the following problem: Let S be a separable and  metrizable topological space and suppose $C(S)$ the space of continuous function and $C_b(S)$ the space of bounded continuous function. Given a continuous function $\psi:S\rightarrow[1,\infty)$, is it true that for any continuous function $f\in C(S)$ with $|f(x)|\leq c\cdot \psi(x), \text{for all } x\in S$, there exist a bounded continuous function $g(x)\in C_b(S)$ such that $f(x)=g(x)\cdot\psi(x)$. If not, under which condition would this hold? The point is that I need this result to state a bijection between the space of $f(x)$ and $C_b(S)$. Any hint is appreciated.

A:

Since $\psi(x) \neq 0$ we may consider $g(x) = \frac{f(x)}{\psi(x)}$. Then $|g(x)| \leq c$ by your inequality, hence $g$ is bounded. And of course $g$ is also continuous. 

