Let A be a non-empty set in a metric space (X,d). Define$\displaystyle f : X \rightarrow \mathbb{R}$ by $\displaystyle f(x) = inf \{d(a,x) : a \in A \}$
. Prove that f is continuous.
I dont see how to bound f(x) and f(y)
Let A be a non-empty set in a metric space (X,d). Define$\displaystyle f : X \rightarrow \mathbb{R}$ by $\displaystyle f(x) = inf \{d(a,x) : a \in A \}$
. Prove that f is continuous.
I dont see how to bound f(x) and f(y)
Hint The class of subsets of $\displaystyle \mathbb{R}$ in the form $\displaystyle G_{\alpha}=(\alpha,+\infty)$ or $\displaystyle H_{\beta}=(-\infty,\beta)$ is a subbasis of $\displaystyle (\mathbb{R},T_u)$ . Prove that $\displaystyle f^{-1}(G_{\alpha})$ and $\displaystyle f^{-1}(H_{\beta})$ are open sets.
Suppose that $\displaystyle a\in A$.
$\displaystyle f(x)=\inf\{d(x,t):t\in A\}\le d(x,a)$
$\displaystyle d(a,x)\le d(a,y)+d(y,x)$ or $\displaystyle d(a,x)-d(y,x)\le d(a,y)$ for each $\displaystyle a\in A$
Thus $\displaystyle f(x)-d(x,y)\le f(y)$ or $\displaystyle f(x)-f(y)\le d(x,y)$
Can you continue to prove that $\displaystyle |f(x)-f(y)|\le d(x,y)~?$