Continuity in Metric Spaces

Theorem: Continuity at a Point

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.

A function $f:X\to Y$ is ^continuity-at-a-point at $x_0\in X$ if and only if for each open ball $B_{\epsilon }(f(x_0))$ around $f(x_0)$ there exists some open ball $B_{\delta }(x_0)$ around $x_0$ such that if $x$ is inside $B_{\delta }(x_0)$, then $f(x)$ is inside $B_{\epsilon }(f(x_0))$.

$$\forall \epsilon >0,\exists \delta >0:x\in B_{\delta }(x_0)⟹f(x)\in B_{\epsilon }(f(x_0))$$

PROOF