Let $(X,d_{disc})$ be a discrete metric space, where $d_{disc}(x,x')=\begin{cases} 1 & \text{if } x\neq x', \\ 0 & \text{if }x=x'.\end{cases}$
Let $(Y,d_Y)$ be an arbitrary metric space, and $\mathbb{R}$ be equipped with $d_{\mathbb{R}}(r,r')=|r-r'|.$
- Prove that any function $f:X\to \mathbb{R}$ is continuous.
- Prove that $d_Y:Y\times Y\to \mathbb{R}$ is continuous. Here articulate which metric you use for $Y\times Y$.
My Idea:
- Observe that $f$ is continuous iff for all $\epsilon>0$, $\exists \delta>0$ such that $d_{\mathbb{R}}(r,r')<\epsilon$ if $d_{disc}(x,x')<\delta.$ Pick $\delta=1/2$ then if $x=x'$ trivially $d_{\mathbb{R}}(r,r')<\epsilon$ or else if $x\neq x'$ then $d_{\mathbb{R}}(r,r')=|r-r'|<\epsilon$? I don't understand how to finish it. Alternatively, as Nate said I could look at the other definition, $f$ is continuous iff forall open sets $U\subset \mathbb{R}$ , $f^{-1}(U)\underset{open}{\subset} X.$ We are concerned with all the open balls with radius less than $1$.
- I am not sure how to proceed on this one. Any comments and hints will be appreciated.Thanks.