Suppose we have an autonomous first-order ordinary differential equation
$$\frac{dx}{dt} = f(x) \tag{*}$$
where $f$ is continuously differentiable for all $x \in D \subseteq \mathbb R$ s.t. the ODE has a unique maximal solution for every initial value problem in $D$.
Book says:
Theorem: Let $u(t), t \in I \subseteq \mathbb R$ be a solution of $(*)$. If $u'(t_0) = 0$ for some $t_0 \in I$, then $u(t), t \in I$ is a constant solution.
Right after that there is:
Corollary: A solution of $(*)$ on some interval $I \subseteq \mathbb R$ is nondecreasing or nonincreasing.
The theorem seems to be saying if the derivative is zero for some value in the interval, then the derivative is zero for the whole interval.
Can we similarly show that if the derivative is positive/negative for some value in the interval, then the derivative is positive/negative for the whole interval?
There's no proof for the corollary given, but I think the proof for the theorem can be modified for 'increasing' instead of 'constant' and then 'decreasing' instead of 'constant'.
Indeed, the theorem is "saying if the derivative is zero for some value in the interval, then the derivative is zero for the whole interval".
Also, indeed one can show that "if the derivative is positive/negative for some value in the interval, then the derivative is positive/negative for the whole interval". The reason is that, by continuity (recall that solutions are supposed to be $C^1$ and so their derivative is continuous), if there were some positive value and some negative value of the derivative, there would exist a point with zero derivative, and thus all should have zero derivative (which contradicts to the hypothesis that there were some positive value and some negative value of the derivative).
Of course, one could also assume that there were some nonzero value and some zero value of the derivative, but according to the theorem this would also lead to a contradiction since all should have zero derivative. In other words, it is safe to consider only the case in the last paragraph: there were some positive value and some negative value of the derivative.