I have a real function that satisfies:
- $f:\mathbb R\rightarrow\mathbb R$ is differentiable at $x$ for $x\neq x_0$.
- There is a full-measure set $T$ such that for any sequence $t_n\in T$ with $t_n\stackrel{n\rightarrow\infty}{\longrightarrow} x_0$ we have $\lim_{n\rightarrow\infty} f^\prime (t_n) = c$.
- $f$ is non-decreasing
I'm trying to show that $\mathbf{f}$ must also be differentiable at $\mathbf{x_0}$.
My idea is that if there is some sequence $x_n$ with $x_n\stackrel{n\rightarrow\infty}{\longrightarrow} x_0$ and $\lim_{n\rightarrow\infty}\frac{f(x_n)-f(x_0)} {x_n-x_0} \neq c$ then I can approximate the $x_n$ and $x_0$ with points in $T$ but I have to be careful with the choice of $\varepsilon$ and $\delta$ in approximation because we have two limits here (the derivative itself and the limit of the derivatives). I'm not sure if this would work because it seems that I would need to change the order of limits. Besides, in real analysis, there are some many pathological cases that perhaps there is even a counter-example. Any suggestions on how to attack this problem?