A uniformly convergent series of continuous functions has a continuous sum.Suggest a slogan
Let \((X,d)\) be a metric space, let \(f_n\colon X\to\CC\) be continuous for every \(n\geq 1\), and let \(F\colon X\to\CC\). If \[ \norm{\sum_{n=1}^N f_n - F}_\infty \coloneqq \sup_{x\in X}\abs{\sum_{n=1}^N f_n(x) - F(x)} \xrightarrow{N\to\infty} 0, \] then \(F\) is continuous.
The partial sums \(S_N\coloneqq\sum_{n=1}^N f_n\) are continuous and converge uniformly to \(F\). Fix \(x_0\in X\) and \(\varepsilon>0\), choose \(N\) with \(\norm{S_N-F}_\infty<\varepsilon/3\), and choose \(\delta>0\) with \(\abs{S_N(x)-S_N(x_0)}<\varepsilon/3\) whenever \(d(x,x_0)<\delta\). For such \(x\), the triangle inequality gives \(\abs{F(x)-F(x_0)}<\varepsilon\).