Prove that $f\notin\text{RS}_{f}([0,1])$.

Question

If $f(x)=\begin{cases} \sqrt{x}\sin{(1/x)},&\hbox{if $x\neq0$}\\ 0,&\hbox{if $x=0$} \end{cases}$

Prove that $f\notin\text{RS}_{f}([0,1])$. I think this can be solved using Reduction to Riemann Integral Theorem, but I don't have any idea. Please any suggestion.

Answer 1

Hint

Denote $x_n = \frac{1}{n\pi + \frac{\pi}{2}}$ and consider the partition

$$0 < x_{2n} < x_{2n-1} < x_{2n-2} < \dots < x_2 < x_1 < 1.$$

Then prove that

$$\begin{aligned}\sum_{i=0}^{2n} f(x_{i+1})\left(f(x_{i+1}) - f(x_i)\right) \end{aligned}$$ diverges as $n \to \infty$.