continuity of series of functions

Question

Given that $f_1:[0,1] \to \mathbb{R}$

where $f_1=\sum_{n=1}^{\infty}\frac{x\sin(n^{2}x)}{n^2}$

The question is, i have to check the continuity of the given series of functions.but i have no idea how to solve further,should i check its uniform convergence first?please give me a hint.

please help

Thank you

Do you know the Weierstrass M-test? https://en.wikipedia.org/wiki/Weierstrass_M-test — BenediktK, Jun 09 '19 at 10:18
$f_1=\sum_{n=1}^{\infty}\left | \frac{x\sin(n^{2}x)}{n^2}\right|$ < $\frac{1}{n^2}$ and $\sum_{n=1}^{\infty}\frac{1}{n^2}$ is convergent so the given function conveges uniformaly — TheStudent, Jun 09 '19 at 10:30

score 0 · Answer 1 · answered May 06 '20 at 19:12

The series converges uniformly to $f_1$ .

Let us define:- $ S_m : [0,1]\to \mathbb {R}$ such that $$ S_m (x)=\sum_{n=1}^{R}\frac{x\sin(n^{2}x)}{n^2} \text{ ; }\forall m\ge 1$$

Observe that each of $S_r$ is a continuous function on $[0,1]$.

It is clear by Weierstrass M-test that $S_r\to f_1$ uniformly on $[0,1]$.

Fact:-Uniform limit of a continuous function is continuous.

Hence $f_1$ is continuous.

Now continuous function on a compact set is uniformly continuous(proof can be found here). And we have $[0,1]$ which is compact.

$\therefore f_1$ is uniformly continuous on $[0,1]$.

continuity of series of functions

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