The total variation of differentiable function $f$ is defined by $$||f||_V=\int_{-\infty}^{\infty} |f'(t)|\:dt$$
I must show $$||f||_V = \infty$$ if $$f(t)=\frac{\sin(\pi t)}{\pi t}$$
In other words I need to solve $$\int_{-\infty}^{\infty} \Big|\frac{\pi t\cos(\pi t)-\sin(\pi t)}{\pi t^2}\Big|\: dt$$ $$2\int_0^{\infty}\Big|\frac{t\cos(t)-\sin(t)}{t^2}\Big|\:dt$$ and here I don't know where to proceed
Hint: in the interval $(2n\pi-\frac {\pi } 4,2n\pi+\frac {\pi } 4)$ verify that $|t \cos t -\sin t| >\frac {t-1} {\sqrt 2}$. Now integrate over this interval and sum over $n$.