$f$ continuous on $[a,b]$ with one-sided Dini derivative is increasing
Solution 1:
Let me be completely perverse and give you no hints for solving this. But after you succeed (and indeed you will) you will then need to know the background and history of this.
First of all some oblique advice. The problem suggests "one of the Dini derivatives" and you chose the right, lower Dini derivative. That makes the problem way too easy! It is far more interesting that this result is true for the upper right Dini derivative. There is a difference!
A bit of History.
In 1878 the Italian mathematician Dini (see reference [1]) proved this theorem, a bit more ambitious than your exercise.
Theorem Suppose that $f:\mathbb R\to\mathbb R$ satisfies
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$\limsup_{h\to 0+} f(x-h) \leq f(x) \leq \limsup_{h\to 0+} f(x+h)$
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The upper right Dini derivative of $f$ is nonnegative at every point with the exception of an at most countable set of points.
Then $f$ is monotone nondecreasing.
If you find this admirable then you will definitely appreciate the extension due to the great Zygmund.
Theorem Suppose that $f:\mathbb R\to\mathbb R$ satisfies
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$\limsup_{h\to 0+} f(x-h) \leq f(x) \leq \limsup_{h\to 0+} f(x+h)$
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The set of values of $f(x)$ at points where the upper right Dini derivative of $f$ is zero or negative contains no interval.
Then $f$ is monotone nondecreasing.
Your resource for this (and a host of other interesting aspects of real analysis) is the monograph of Saks [2], pages 203--207.
If you find yourself short of ideas to complete the assignment learn the techniques here. The Saks book is available for free online or also it can be found in a paperback version. It represents what was known in the 1930s and you can be sure that it takes some time for students to get even that far in understanding properties of real functions.
REFERENCES.
[1] Dini, U. (1878). Fondamenti per la teorica delle funzioni di variabili reali. Pisa: Tipografia T. Nistri e Co.
[2] S. Sak, Theory of the Integral, 1937 https://archive.org/details/theoryoftheinteg032192mbp/page/n9/mode/2up