Given the function $F(x)=\frac{\pi}{2}\sin(x)$. Find the fixed points and, if they exist, determine if the points are attractors or repulsors without differentiation.
I already found the fixed points but I need help in the second part.
My doubt is if there is any way to determinate if the points are attractors or repulsors without using differentiation (In the course that I’m in, I can't use differentiation yet).
For a fixed point $x$, consider a point $x+\varepsilon$ close to $x$ and try to determine whether the map $x+\varepsilon\mapsto \frac{\pi}{2}\sin(x+\varepsilon)$ takes you nearer to $x$ or further from it. For instance, if $x=0$, then $\epsilon\mapsto\frac{\pi}{2}\sin\epsilon\approx\frac{\pi}{2}\epsilon$, which is further away from $0$. So $0$ is a repulsor.