$f(x) = \sin\frac{1}{x}+2x\cos\frac{1}{x}$ is not of bounded variation

Question

Let $f: [-1,1] \rightarrow \Bbb{R}$ such that

$f(x) = \sin\frac{1}{x}+2x\cos\frac{1}{x}$ for $x$ not equal to $0$

and $f(0) = 0$.

I need to show that $f$ is not of bounded variation.

I have shown that $g(x)$ = $\sin\frac{1}{x}$ for $x$ not equal to $0$ and $g(0) =0$ and $h(x)$ = $2x\cos\frac{1}{x}$ for $x$ not $0$ and $0$ for $x$ = $0$ are not of bounded variation.

But I don't think I can say anything about $f$ from here?

Help, please.

Answer 1

$f(x)$ is not bounded because $\lim_{x \to 0} \sin(1/x)$ does not exist, because if you Take $x_n=\frac{1}{n\pi}, x_n'=\frac{1}{(n+1/2)\pi}$, then $\sin n\pi \ne \sin (n+1/2) \pi$