# Continuity in Metric Spaces

• October 30th 2012, 08:21 AM
I-Think
Continuity in Metric Spaces
Just need to check the details in my proof are correct
Question
$(X,d)$ is a metric space, fix $x_0\in{X}$, let $F_{x_0}=d(x_0,x)$. Show this function is continuous

Proof
Consider the open set $W\in{\mathbb{R}}$ and let
$F_{x_0}^{-1}(W)=[x\in{X}|F_{x_0}(x)=d(x_0,x)\in{W}}$]. For all $y\in{W}$, $\exists{\epsilon_y}>0$ such that $B(y,\epsilon_y)\subset(W)$
Let $z=F_{x_0}^{-1}$ so $z\in{F_{x_0}^{-1}}$, consider $B(z,\epsilon_y)$

Let $\alpha{\in{B(z,\epsilon_y)}$, $F_{x_0}^{-1}(\alpha)$ $=d(x_0,\alpha)
So $F_{x_0}^{-1}(\alpha)\in{B(y,\epsilon_y)}\subset{W}$, so $B(z,\epsilon_y)\subset{F_{x_0}^{-1}(W)}$
So ${F_{x_0}^{-1}(W)$ is open and the result follows