Prove that the function $\displaystyle f(x)=\sum^{\infty}_{n=1} |\sin x|^{\sqrt{n}}$ is continuous on $\displaystyle (-1,1)$. Examine differentiability of it on $\displaystyle (-1,1)$.
Yes (maybe you should detail the fact that the series $\displaystyle \sum_{n=0}^{+\infty}a^{\sqrt n}$ converges for $\displaystyle 0<a<1$). Use the fact that the series is normally convergent. For the differentiability, look at $\displaystyle 0$.