Show that this function is invertible Show that 
$$
f(x)=\frac{1}2\sin(2x) + x
$$
is invertible.
How do I continue with this? 
I've tried with taking the derivative and taken the fact that:
$$
f'(x)=\cos(2x)+1 \geq 0
$$
Is this enough? I'm not sure since it can also equal $0$...

Depends how fussy you are. Suppose that $a\lt b$. We want to show that $f(a)\lt f(b)$. 
Step a tiny amount to the right of $a$, say to $c$, where $c\lt b$ and there is no $x$ strictly between $a$ and $c$ such that $f'(x)=0$. Then $f(a)\lt f(c)$. Also, $f(c)\le f(b)$, so $f(a)\lt f(b)$.
