# What is an example of a non-convex region?

In complex analysis, the proof of Morera's Theorem

"for $f\in C(D)$ such that D is a region, if for any triangle $\triangle$ in D, $\int_{\triangle}f = 0$ is True, then f is analytic in D."

splits the proof into two cases: for convex and non-convex regions D.

I'd like some intuition before my exam on what a non-convex region might look like. Can you provide an example?

EDIT:

I apologize gentlemen, this question has shown me that the definition of a region is regrettably localized by textbook.

In Bak-Newman, a region is defined as an open, connected set in $\mathbb C$, which would make @Did's examples valid if considered with their boundaries removed in $\mathbb C$.

• Did's examples are invalid? Can you not imagine a banana-shaped open connected set in $\mathbb C$? – user856 Oct 18 '13 at 0:59
• Yes, I understand how to extend his examples to satisfy my definition. I'd like him to change his answer to suit my edit so I can accept it. – Jake Color Oct 18 '13 at 1:03

A banana. A necklace. A tire. An eight (actually no Roman digit is convex except "one" when it is written as a vertical bar instead of as $1$).
Something with a hole in the middle is non-convex, but can still be connected. In particular, $\mathbb{C} \setminus \{0\}$.