The right-hand derivative of a certain integral function (Q1332605)

Given a strictly decreasing null sequence in \([0,1]\) and \(A\in \mathbb{R}\), two bounded functions \(f_ i: (0,1]\to \mathbb{R}\) are exhibited such that \(f_ i\) are continuous on \((0,1]\), \(f^{-1}_ i(0)= \{x_ n: n\in \mathbb{N}\}\), \(f_ i(0+)\) does not exist, \(F_ 1'(0)= A\), and \(F_ 2\) is not differentiable at 0, where \(F_ i(x)= \int^ x_ 0 f_ i\), \(i\in \{1,2\}\).