Given two metric spaces \((X,d_X)\) and \((Y,d_Y)\). A function \(f\colon X\to Y\) is continuous in \(x\in X \) if and only if \(x_n\to x\) implies \(f(x_n)\to f(x)\).
Proof
Apply continuity directly in the forward direction, to ensure \(d(f(x_n),f(x))<1/k\) for all \(n>N_k\).
For the reverse direction assume \(f\) is not continuous and construct a sequence \(x_n\to x\) such that \(f(x_n)\nrightarrow f(x)\).
Remarks
- For the prove we can use that \(X\) is first countable . Then this proposition is a consequence of (0x677cfb01) and (0x6770f781) .