I was asked to prove or disprove this claim. I've tried to find a counter example but I've got a feeling now that the statement is correct. My guess is they want us to prove it using the squeeze rule, but how would I approach this? I can see straight away that $0 < f(x) \le1$, but how does that help me? should I perhaps try to use the contrapositive or proof by contradiction?
Hints
Does $f(x)$ have to be a positive function?
Can $f(x)$ be positive for $x < 0$?
Can $f(x)$ have two specifications, one for positive $x$ and one for negative $x$?
Does $g(x)$ have to be a positive function?
Can you define a function $f(x)$ such that $xf(x)$ goes towards $-\infty$ as $x$ approaches $0$ from above?