Continuity of series

Question

Prove that $\displaystyle \sum_{x=1}^{\infty} \frac{1}{x^p}$ is continuous for p>1.

I can prove that this series converges uniformly. However, does this imply continuity? How do I prove continuity for a series?

Answer 1

Yes. If a sequence of continuous functions is uniformly convergent, the limit is continuous. Proof is a classical ecample of "$\varepsilon/3$" argument:

Proof: Let $f_n$ be sequence of continuous functions and $f_n\to f$ uniformly, say, on $[a,b]$. Let $\varepsilon>0$, $x_0\in(a,b)$. By uniform convergence, for sufficiently large $n$ we have $\|f-f_n\|<\varepsilon/3$. Since $f_n$ is continuous, there exists a neighbourhood $U$ of $x_0$ such that $\left|f_n(x)-f_n(x_0)\right|<\varepsilon/3$ for $x\in U$. Combining all of these, for $x\in U$ we have $$ \left|f(x)-f(x_0)\right|\leq \left|f(x)-f_n(x)\right|+\left|f_n(x)-f_n(x_0)\right|+\left|f_n(x_0)-f(x_0)\right| \leq\frac{\varepsilon}{3}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}= \varepsilon $$ hence $f$ is continuous at $x_0$.