Existence of a continuous function $\rho(x)$ such that $g(x) \rho(x)$ is bounded and continuous for $g(x)$ continuous

Question

How can I precisely prove the existence of a continuous function $\rho(x)$ such that $0 \leq \rho(x) \leq 1 \forall x \in R^d $ such that $g(x) \rho(x)$ is bounded and continuous for $g(x)$ continuous?Both $g(x)$ and $\rho(x)$ are defined on $R^d$.

My idea was that we can choose $\rho(x)$ such that $\rho(x)g(x)$ goes exponentially to zero outside a compact set in $R^d$.But i cant argue rigorously?

Can hints on how could I proceed?

Answer 1

How about $$ \rho(x)=\frac{1}{1+|g(x)|}? $$

Answer 2

How about:

$$\rho(x)\equiv 0$$