Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, let $f:X\to Y$ be a function, and let $x_0\in X$. For $a\in X$ and $r>0$, write $B_X(a,r)=\{x\in X:d_X(x,a)<r\}$, and for $b\in Y$ and $s>0$, write $B_Y(b,s)=\{y\in Y:d_Y(y,b)<s\}$. Then $f$ is continuous at $x_0$ if and only if for every $\varepsilon>0$ there exists $\delta>0$ such that

\begin{align*} f(B_X(x_0,\delta))\subset B_Y(f(x_0),\varepsilon). \end{align*}