Prove that $T$ is a homeomorphism of the circle if $|\epsilon|<1$

Question

can someone give a help on this? Let $T:S^1 \rightarrow S^1$ be given by $T(\theta) = \theta + \omega + \epsilon \sin (\theta)$ where $\omega$ and $\epsilon$ are constants. Prove that $T$ is a homeomorphism of the circle if $|\epsilon|<1$.

Answer 1

The only problem you might have is that $T$ is not one-to-one, since it is clearly continuous and periodic. But $$ T'(\theta)=1+\epsilon\cos\theta $$ which is strictly positive if $|\epsilon|<1$. So under this condition it is a homeomorphism.