Problem:
Let $f: [1,\infty) \to \mathbb{R}$ continuous and $\sup_{x \geq0} x^2|f(x)| \lt \infty$ holds.
Show, that the improper integral $\int_1^\infty f(x) dx$ exists and
$$\int_1^\infty f(x)dx = \int_0^1 \frac{1}{t^2}f(\frac{1}{t}) dt $$ holds.
I need to know, how to obtain the improper integral from the bounded function and out of the supremum. Is it manageable, to get $f$ into the form necessary to say that it's bounded, nothing is known about the function other than it's continuous. It would be nice if I get to see the proof of this.