If the continuous partial sums converge uniformly, the sum is continuous.Suggest a slogan
Let \((X,d)\) be a metric space and let \(f_n\colon X\to\CC\) be continuous for every \(n\geq 1\). If the partial sums \(S_N\coloneqq\sum_{n=1}^N f_n\) converge uniformly on \(X\) to a function \(f\colon X\to\CC\), then \(f\) is continuous.
Each \(S_N\) is a finite sum of continuous functions, hence continuous. Fix \(x_0\in X\) and \(\varepsilon>0\). Choose \(N\) with \(\abs{S_N(x)-f(x)}<\varepsilon/3\) for all \(x\in X\), and then \(\delta>0\) with \(\abs{S_N(x)-S_N(x_0)}<\varepsilon/3\) whenever \(d(x,x_0)<\delta\). For such \(x\), \[ \abs{f(x)-f(x_0)}\leq\abs{f(x)-S_N(x)}+\abs{S_N(x)-S_N(x_0)}+\abs{S_N(x_0)-f(x_0)}<\varepsilon. \]