Equivalent definitions of continuity (metric spaces)

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces , and let $f:X\to Y$.

Fix a point $x_0\in X$. The following are equivalent:

• Epsilon–delta continuity at $x_0$: for every $\varepsilon>0$ there exists $\delta>0$ such that $d_X(x,x_0)<\delta \implies d_Y(f(x),f(x_0))<\varepsilon.$
• Sequential continuity at $x_0$: for every sequence $(x_n)$ in $X$, $x_n\to x_0 \implies f(x_n)\to f(x_0).$
• Neighborhood formulation: for every open set $V\subseteq Y$ with $f(x_0)\in V$, there exists $\delta>0$ such that $B_X(x_0,\delta)\subseteq f^{-1}(V).$

Moreover, $f$ is continuous on all of $X$ if and only if preimages of open sets are open: $V\subseteq Y \text{ open} \implies f^{-1}(V)\subseteq X \text{ open}.$

These equivalences let you choose the most convenient continuity definition for a given proof.

Proof sketch: Epsilon–delta $\Rightarrow$ sequential is immediate by applying the epsilon–delta condition to the tail of a convergent sequence. Sequential $\Rightarrow$ epsilon–delta is proved by contraposition: if epsilon–delta fails, build a sequence $x_n\to x_0$ with $f(x_n)$ staying a fixed distance from $f(x_0)$. The open-set formulation is obtained by taking $V$ to be an open ball around $f(x_0)$, and conversely by applying epsilon–delta to those balls.