It seems obvious that in an arbitrary normed space $(X, \|\cdot\|)$ a mapping $T$ defined as
$ T(x) = \begin{cases} \frac{x}{\|x\|} & t \neq 0 \\\\ 0 & t = 0 \end{cases} $
is continuous everywhere but 0. However, I'm having difficulty formulating a proof. It's quite simple to see that $\|T(x) - T(y)\| \leq 2 \; \forall x,y \neq 0$ and that $\|T(x)\| = 1 \; \forall x \neq 0$, but those properties don't seem to be helpful so far. Where I seem to be having trouble is in finding the right $\delta$ so that $\|x - x_0\| < \delta$ can be brought to bear on $\|T(x) - T(x_0)\|$.
Thanks for any guidance.