Maximum principle and a differential inequality

Question

Let $z:[0,T]\to \mathbb{R}$ satisfy the following differential inequality$$ \frac{d}{dt}z(t)\leq f(z(t)) $$ where $f$ is a continuous real-valued function. Now, let $y$ be a solution to the equality case, i.e, a solution to the ODE$$ \frac{d}{dt}y(t)=f(y(t)). $$ Assuming $x(0)\leq y(0)$, my goal is to apply the maximum principle to conclude that$$ z(t)\leq y(t) $$ for all $t$. So far, I have been unable to conclude anything due to the generality of $f$, but I might be missing something. Any ideas?

Answer 1

This is not a proof, just a sketch of how to think about the problem:

Try looking at the problem in the phase plane: $z$ on the $x$ axis and $dz/dt$ on the $y$ axis. Plot $f(z)$ on the graph and realize that positive points higher on the graph move to the right faster than positive points lower on the graph. Similarly, negative points very low move to the left faster than negative points higher up. Thus, any initial point that starts below $f(z)$ will move right slower than a point on $f(z)$ and move left faster than a point on $f(z)$. Thus, any initial point $z(0)\le y(0)$ must satisfy $z(t)\le y(t)$.