The question is on Page 115, the 18th exercise about Lebesgue Differential Theorem.
Suppose $f$ is a continuous function on $[a,b]$, $\forall x \in [a,b]$, we have $\max\{D^+(f)(x),D^-(f)(x)\} \geq 0$.(Here $D^+,D^-$ denote Dini derivative.) Prove that $f$ is monotone on $[a,b]$.
I have proved the case when only one side derivative is positive. But I can't deal with the case about both sides.
I just realized I confused the Dini derivatives: the method I wrote below would only work for $D_+$ and $D_-$, but the OP reads $D^+$ and $D^-$. I am leaving the method, for archeological purposes.
By contradiction if there is $a'<b'$ with $f(a')>f(b')$ construct $a_n$ and $b_n$ with $a_0=a'$, $b_0=b'$ and: consider any $c_n\in(a_n,b_n)$ such that $f(c_n)$ is the middle of $f(a_n)$, $f(b_n)$. If $|c_n-a_n|\leq |c_n-b_n|$ then let $b_{n+1} = c_n$ and $a_{n+1}=a_n+\epsilon_n$ where $\epsilon_n$ is chosen very small. Proceed similarly if $|c_n-a_n|> |c_n-b_n|$. Then I expect that the common limit of $a_n$, $b_n$ and $c_n$ should be a point $x$ where the hypothesis does not hold. Provided you choose $\epsilon_n$ carefully at each step.