The problem is from Richard's Foundations of Mathematical Analysis problem 42.11 The full problem description is the following
Let $f$ be a function on $[a, b]$. Let $K$ be a compact subset of $[a, b]$ on which $f$ is continuous. Suppose there exists $c > 0$ such that for each $x \in K$, there exists $h_x > 0$ with $$ |\frac{f(x + h_x) - f(x)}{h_x}| < c $$ Prove that there exists a finite subset $\{x_1, ... , x_n\} \subset K$ and positive numbers $h_1, ..., h_n$ such that
a) $x_1 < x_1 + h_1 \leq x_2 < x_2 + h_2 \leq x_3 < ...$
b) $|\frac{f(x_i + h_i)}{h_i}| < c$ for $i = 1, ..., n$
c) $K \subset \bigcup_{i = i}^{n}[x_i, x_i + h_i]$
Here is what I have tried:
I first tried to understand the question and get some intuitive understanding. I failed. I do not really understand what the question is trying to ask behind the scene. But I still had some thought: the question mentioned that finite subset also compact set, I think there might be a connection. Also, I found this is actually very similar to a change rate on $h_x$, thus I did the following: $$b = a + h_a$$, thus we got $$|\frac{f(b) - f(a)}{b - a}| < c$$, but I still do not know how this related to finding a "monotone" subset of the domain.
I also asked my friend who solved the problem, he used a way that involving $$y_x = x + h_x$$, and $$g_x(a) = |\frac{f(y_x) - f(a)}{y_x - a}|$$. I felt like he put it in a compound function and $g_x$ seemed like a partial change rate to me.
Here is what I am looking for help:
- I did not really understand this question intuitively. Can anyone explain where is this problem "evolved" from, or what is a similar problem from which we can drive this question?
- I would love you guys provide me some intuitive way to think of the question instead of a rigorous proof (which, of course, also welcome)
- Can provide me some similar problems and how I should crack such problem from the beggining.
Thank you very much for your help!!!!