Let $A: \mathbb R_+\to [0,1]$ be $1/2$-Holder continuous and $k: \{(s,t): 0\le s\le t\}\to\mathbb R$ be continuous. Define $f:\mathbb R_+\to\mathbb R$ by

$$f(t):=\int_0^t\frac{k(s,t)}{\sqrt{t-s}}\big(A(t)-A(s)\big) \, ds, \quad \forall t\ge 0.$$

Under which condition $f$ is differentiable on $(0,\infty)$? The same question is asked for $g$ that is defined as

$$g(t):=\int_0^tK(s,t)\big(A(t)-A(s)\big) \, ds,\quad \forall t\ge 0,$$

where $K: \{(s,t): 0\le s< t\}\to\mathbb R$ is continuous s.t. $|K(s,t)|\le C/\sqrt{t-s}$ for some $C>0$.