MathDB

Let

f:[0,1]\to\mathbb{R}

be a continuous function. Prove that

\lim_{n\to\infty}\int_0^1 f(x^n) \ dx=f(0).

Furthermore, if

f(0)=0

and

f

is right-differentiable in

0{}

, prove that the limits \lim_{\varepsilon\to0}\int_\varepsilon^1\frac{f(x)}{x} \ dx \text{and} \lim_{n\to\infty}\left(n\int_0^1f(x^n) \ dx\right)exist, are finite and are equal.

Problem Statement