real analysis - Show that a certain function is continuous

Suppose that $S^1$ is the unit circle in $\mathbb{C}$ and suppose that $g\colon [0,2\pi]\rightarrow \mathbb{C}$ is continuous such that $g(2\pi) = g(0)$.

I have to show that $h\colon S^1 \rightarrow \mathbb{C}$, $x \mapsto g(t(x))$ is continuous. where $t(x)$ is the number such that $x = e^{it(x)}$.

I guess that I have to show that if $y \in B(x,\epsilon)$, then $y \in B(t(x),\epsilon)$ (if $\epsilon$ is small enough). This directly implies the result. Can anyone help me to show this?