Let $\{f_n\}$ be a sequence of real-valued $C^1$ functions on [0,1] such that for all n, $|f'_n(x)|\le 1/\sqrt x$
($0<x\le 1$) and $\int_{0}^{1}f_n(x)dx=0$

prove that $\{f_n(x)\}$ has a subsequence which converges uniformly on [0,1]

I think that I should apply arzela-ascoli theorem here, but I don't know how to start. How can i show equicontinuity and uniform boundedness of these functions?


Equicontinuity at every point of $(0,1]$ is clear from the Mean Value Theorem. Hint for equicontinuity at $0$: For every $\epsilon>0$ there exists $\delta>0$ such that $$\int_0^\delta\frac{dx}{\sqrt x}<\epsilon.$$

Hint for pointwise boundedness: If $f$ is $C^1$ and $\int_0^1f=0$ then $$f(x)=f(x)-\int_0^1f(t)\,dt=\int_0^1(f(x)-f(t))\,dt,$$so$$|f(x)|\le\int_0^1\int_x^t|f'(s)|\,dsdt.$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.