# Thread: differentiability and continuity of function

1. ## differentiability and continuity of function

Prove that the function $f(x)=\sum^{\infty}_{n=1} |\sin x|^{\sqrt{n}}$ is continuous on $(-1,1)$. Examine differentiability of it on $(-1,1)$.

2. ## Re: differentiability and continuity of function

What did you try? Did you already show the convergence of the series?

3. ## Re: differentiability and continuity of function

Because of $|\sin x|<\sin 1<1$ on $(-1,1)$we know that the series is convergent. Is that right?

4. ## Re: differentiability and continuity of function

Yes (maybe you should detail the fact that the series $\sum_{n=0}^{+\infty}a^{\sqrt n}$ converges for $0). Use the fact that the series is normally convergent. For the differentiability, look at $0$.