On Euclidean subspaces, continuity is equivalently topological, epsilon--delta, sequential, or open-preimage continuity.Suggest a slogan
Let \(E\subseteq\RR^n\) and let \(f\colon E\to\RR^m\). The following are equivalent:
\(f\) is continuous on \(E\), with the subspace topology on \(E\).
For every \(x\in E\) and every \(\varepsilon>0\) there exists \(\delta>0\) such that \(\norm{f(y)-f(x)}<\varepsilon\) for all \(y\in E\) with \(\norm{x-y}<\delta\).
For every sequence \((x_n)_{n\geq1}\) in \(E\) converging to a point \(x\in E\), \(\lim_{n\to\infty}f(x_n)=f(x)\).
For every open set \(G\subseteq\RR^m\), the set \(\inverseof{f}(G)\) is open in \(E\) with the subspace topology. [@Rud76].