Proving continuity of a function defined as a series of functions. How can I prove that the real valued function $$ f(x)=\sum_{n\geq 1} \sqrt{x} e^{-n^2 x} $$ is continuous for $x>0$ ? I have tried to use the Weierstrass $M$-test to prove the uniform convergence of the series of functions and from the uniform limit theorem conclude the proof, but I can't find the bound to use the $M$-test. This question is from the book Introduction to Calculus by Kazimierz Kuratowski, and is presented before derivatives and integrals, so I think it's pretended to be solved without that.

For any $\delta>0$, you can use "$M$-test" on $[\delta, \infty)$: say $$ \sqrt x e^{-n^2x}=\sqrt x e^{-n^2x/2}e^{-n^2x/2}\leq C\cdot e^{-n^2\delta/2}\leq C\cdot e^{-n\delta/2} , $$ where $C=\sup_{x\in[0, \infty)}\sqrt x e^{-x/2}$. So $f$ is continous on every $[\delta, \infty)$.