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: