Is the primitive of a density function Hölder continuous?

Question

Let $I=[-1,1]$ and let $f\in L^1(I)$.

Is it true that the function $F:I\to\mathbb{R}^+$ defined as $$F(s)=\int_{-1}^s |f(x)|dx$$ belongs to $C^{0,\alpha}(I)$ for a certain $\alpha>0$?

Answer 1

For $j=1,2\dots$ let $\delta_j=2^{-j^2}$. Define $$c_j=\delta_j^{-1+\frac1j},$$ $$I_j=(0,\delta_j)$$ and $$f=\sum_1^\infty c_j\chi_{I_j}.$$ Since all the terms are non-negative, for every $\alpha>0$ we have $$F(\delta_j)-F(0)=F(\delta_j)\ge\int_0^{\delta_j}c_j=\delta_j^{1/j}\ne O(\delta_j^\alpha).$$