Determine continuity and differentiability of the real function $f(x)=\sum\limits_{n\geq1}\frac{1}{n^x}$

Question

I've been asked to analyze the domain, continuity and differentiability of the function $$f(x)=\sum\limits_{n\geq1}\frac{1}{n^x}$$ I've already shown that this function is defined for every real greater than 1. Now, I suspect It is continuous and differentiable over It's whole domain, so I'm trying to use the Weierstrass M test to show that this series uniformly converges for every $x>1$. But all I have is that if $x\geq2$ then $\frac{1}{n^x}\leq\frac{1}{n^2}$, which combined with the fact $\sum\limits_{n\geq1}\frac{1}{n^2}=\frac{\pi^2}6<\infty$ results in that $f$ is continuous for $x\geq2$. But I don't know how to deal with numbers smaller than 2. I'd apreciate some hints or even better, some ideas to deal with these kind of problems in a more direct way. Thanks in advance.

Answer 1

This is the Riemann zeta function. Here are some useful things you can read: Click here !