Suppose we have a sequence $f_n:[a,b)\to\mathbb R$ of right differentiable functions (i.e., the right derivative $$ f_{n+}'(x)=\lim_{h\to0^+}\frac{f(x+h)-f(x)}{h} $$ exists for all $x\in[a,b)$.) and the right derivatives $f_{n+}'$ converge uniformly to a function $g:[a,b)\to\mathbb R$. Suppose also that there is $x_0\in[a,b)$ such that $f_n(x_0)$ is a convergent sequence of real numbers. Does this imply that there is a function $f:[a,b)\to\mathbb R$ such that $f_n$ converges uniformly to $f$ and $f_+'=g$?
This is true if we are considering the standart derivative, but I can't see how this could be prooven analogously, one of the reason being that I've always seen the MVT being used on the standard problem, but for lateral derivatives it doesn't hold in general (or at least I don't know about some similar result), as for the function $h:[-1,1)\to\mathbb R$ given by $h(x)=x+1$ if $x<0$ and $h(x)=x-1$ if $x\ge0$.
Am I missing something? Or is there not enough information for this problem?