Continuity and Metric Spaces How do I show that the function $f:X \to \mathbb R$ given by $$f(x)=\frac{d(a,x)}{d(a,b)}$$ is continuous.
Given that $(X,d)$ is a metric space, and $a,b$ are distinct points in $X$.

It is well-known, and easy to prove, that $d:X\times X\to \mathbb R$, the distance function in any metric space, is uniformly continuous. In particular then, the function $d(a,-):X\to \mathbb R$ is continuous, and (since $d(a,b)\ne 0$), so is the function you mention.  
