$\int_0^1 \frac{{f}(x)}{x^p} $ exists and finite $\implies f(0) = 0 $ Need some help with this question please.
Let $f$ be a continuous function and 
let the improper inegral 
$$\int_0^1 \frac{{f}(x)}{x^p} $$ 
exist and be finite for any $ p \geq 1 $.
I need to prove that 
$$f(0) = 0 $$
In this question, I really wanted to use somehow integration by parts and/or the Fundamental theorem of calculus.
Or even maybe use Lagrange Means value theorem,
but couldn't find a way to se it up.
I'll really appreciate your help on proving this.

Hint: Let
$$
g(t) = \int_t^1 \frac{f(x)}{x^p}dx
$$
Now investigate properties of $g(t)$ around $t=0$.
