I'm dealing with a calculus exercise. Let me first explain my notation: I say $f_-^\prime(x)$ exists provided the limit $\lim_{t\to x^-}\frac{f(t)-f(x)}{t-x}$ exists. Also we define four Dini derivatives:
Here is the problem:
Let $f\in C[a,b]$, and $f_-^\prime(x)$ exists for all $x\in(a,b)$. Prove that$$\inf_{x\in(a,b)}\{f_-^\prime(x)\}\le\frac{f(b)-f(a)}{b-a}\le\sup_{x\in(a,b)}\{f_-^\prime(x)\},$$then conclude that if $f_-^\prime(x)$ is continuous on $(a,b)$ with respect to $x$, then $f\in D(a,b)$.
I'm confused with this problem. First of all, my search of internet suggested me the following theorem (here):
therefore it seems that the problem follows from the theorem, since the existance on the interval $(a,b)$ suggests that one-side derivative exists and finite except for possibly two points (the endpoints of an interval), which measures zero. However, I don't think this is the right way to solve this, since I don't think it is proper to use such a profound result on an exercise from my textbook.
Can someone provide an alternative approach to solve this? Thanks for your advice.