A problem in Student Olympiad Contest in Vietnam

Question

Let $f$ is continuous on [0,1] and $f(0)=0$. Prove that: $$\lim_{a\to0^+}\int\limits_{0}^{1}ax^{a-1}f(x)dx=0.$$ I have been pondering this problem for days, I have received advice on using lebesgue theorem but I have not learned anything about measure. I'm stuck with a problem: when a is small enough, the integral becomes a generalized integral and I haven't found a solution yet. Probably the only thing I've discovered is that this integral is like a partial integral but f(x) is not differentiable.

Answer 1

Hint: There exists $\delta >0$ such that $|f(x)|<\epsilon$ for $0\leq x<\delta$. Split the integral into the integral over $[0,\delta]$ and the one over $[\delta, 1]$. In the second part use an upper bounf for $|f|$ and evaluate $\int_{\delta}^{1}a x^{a-1}$ explicitly.