Given two metric spaces \((X,d_X)\) and \((Y,d_Y)\). A function is called continuous in \(x\in X\) if for every \(\varepsilon>0\) there is a \(\delta>0\) such that \(d_X(x,x')<\delta\) implies \(d_Y(f(x),f(x'))<\varepsilon\).
We say \(f\) is continuous if it is continuous in every point.