Define for $f:[0,b]\to\mathbb R$ $$ F(t):=\int_0^tf(t-s)s^{-\gamma}\,ds,\quad \gamma\in(0,1). $$ (i) Does a continuous $f$ exists so that $F$ not differentiable on $(0,b)$?
(ii) Does a continuous $f$ exists so that $F'$ exists on $(0,b)$ but it is not integrable?