# Ngô Quốc Anh

## September 17, 2008

### L^2 differentiable?

Filed under: Các Bài Tập Nhỏ, Giải Tích 2, Giải Tích 3 — Ngô Quốc Anh @ 13:04

Let  and define .

a) Must  be differentiable at 0?
b) Must  have any differentiable points?
c) Let , show that  exists and determine what it is.

Solutions.

a) No. For example, let  for  so that  near zero. This  but  does not exist.

b) Yes. In fact,  must be differentiable almost everywhere, by the Lebesgue theorem on the differentiation of the integral. This theorem requires only that  which is true.

c) By the Schwarz inequality,

$f^2(x) = \left(\int_0^x}g(t)\,dt\right)^2\le \left(\int_0^x 1\,dt\right) \left(\int_0^xg^2(t)\,dt\right).$

At least that’s it for  Being careful about the other side, we determine that

$0\le\phi(x) = f^2(x)\le|x|\left|\int_0^xg^2(t)\,dt\right|.$

But since  is an integrable function we have (by an argument that uses the Dominated Convergence Theorem) that

$\lim_{x\to 0}\int_0^xg^2(t)\,dt = 0.$

Hence $\lim_{x\to0}\frac {\phi(x)}{x} = 0,$ so