# Math Help - help with continuous function proof

1. ## help with continuous function proof

i need help

Let (X,d) be a metric space. Let a be element of X. Define a function f: X -> R by f(x) = d(a,x) show tat f is a continuous function.

this is what I've done

abs( x-y) = d(x,y) leq d(x,a) + d(a,y) by triangle inequality thm < delta = epsilon.

now abs(f(x)-f(y)) = abs(d(a,x)-d(a,y)) leq d(x,a) + d(a,y) < epsilon

does this make any sense???

2. Here is a theorem:
If $\left\{ {a,x,y} \right\} \subseteq X$ then $\left| {d(a,x) - d(a,y)} \right| \leqslant d(x,y)$. Can you prove that?
Your problem follows at once from that theorem.