showing a continuous function is invertible given the following conditions

Question

Question:

Let $f:[0,\infty] \rightarrow [0,\infty]$ be a continuous function such that $f(0)=0$ and $x-y \leq f(x) - f(y)$ whenever $x\geq y \geq 0.$ Show that $f$ is invertible.

Could anyone give me any tips how to think about this and the right way of going about showing this? TIA

Answer 1

Try to prove if $x_1 \le x_2$ and $f(x_1) - f(x_2) = 0$ then $x_1 = x_2$,

Also prove that $f(x) \ge x$ and then use IVT,

That will give $f$ is onto