The Group of Complex Continuous Functions?

Question

Let $C(\mathbb{C},\mathbb{C})=\{f:\mathbb{C} \rightarrow \mathbb{C}\,|\,f $continuous $\}$ be the set of all continuous functions from the complex plane to itself and consider the composition operator: $$ \circ:C(\mathbb{C},\mathbb{C})\times C(\mathbb{C},\mathbb{C}) \rightarrow C(\mathbb{C},\mathbb{C})$$

Is $ ( C(\mathbb{C},\mathbb{C}), \circ)$ a Group?

...I lack some insight about the continuity of the inverse functions.

Is the inverse of a continuous $f:\mathbb{C}\rightarrow\mathbb{C}$ continuous? Proofs?

Answer 1

Functions that are not one-to-one do not have inverses.