Please Help Me:
Let $f:[a,b]\to\mathbb R$ be a continuous map. Then the image points $f(t);t\in[a,b]$ are continuously ordered according to the increasing value of $t.$
I don't understand what is meant by "continuously ordered according to the increasing value of $t$" and how can it be shown?
Of course it doesn't mean $\forall ~x,y,z\in[a,b]$ with $x< y< z,$ $|f(x)-f(y)|$$\leq |f(x)-f(z)|...(1)$ for if we consider $f:[0,2\pi]:t\mapsto \cos 2t+i\sin 2t$ then $|f(0)-f(2\pi)|<|f(0)-f(\pi)|.$ Then does it suggest $\exists$ an interval in which $1$ holds?
Also how can it be guranteed that $Im~f$ always looks like a line (not necessarily straight)?