Question on continuity in Topology $X=[0,2\pi)$ with the relative topology determined by the usual topology on $\mathbb R$, and the unit circle $S=\{(x_1,x_2)\mid x_1^2+x_2^2=1\}$, with the relative topology determined by the usual topology on $\mathbb R^2$. Define $f:X\to S$ by $f(t)=(\cos(t),\sin(t))$. Prove $f$ is continuous and prove $f$ inverse is not continuous. 

I keep looking at this and for some reason solve it. Please help me
HINT: First show that the inverse image of an open interval in $S$ is either an open interval in $X$ or a set of the form $[0,x)\cup(y,2\pi)$, which is also open in $X$; from this you can conclude that $f$ is continuous. Now let $g=f^{-1}$. $[0,1)$ is an open set in $X$; is $g^{-1}\big[[0,1)\big]$ open in $S$?
f^(-1) is not even a function as df is not 1-1.