I'm trying to show:
Let $(X,d)$ be a metric space and let $A, B$ be nonempty subsets, which are also closed and disjoint. Let $\rho_A:X\to \mathbb{R}$ be such that $\rho_A=d(x,A)$ and $\rho_B:X\to \mathbb{R}$ be such that $\rho_B=d(x,B)$, with $x\in X$ (distance from one point to a set).
Prove that the function $\frac{\rho_A}{\rho_A+\rho_B}:X\to \mathbb{R}$ is continuous in $X$.
I have only the definition of continuous function (with balls) and some results. I can not use yet sequences.
A corollary says that composition of continuous functions is a continuous function on metric spaces. Now, the sum of continuous functions is continuous (in the metric space $\mathbb{R}$) but the division of continuous functions is not necessarily a continuous function in $\mathbb{R}$.
Any help? Thank you very much.