Mapping/Mistakes/Image Element Multiply Defined

Example of Mistake in Definition of Mapping
This example of an attempted definition of a mapping contains a mistake.
 * $h: \R \to \R$ defined as: $\forall x \in \R: x \mapsto \begin{cases} x + 1 & : x \ge 0 \\ 0 & : x \le 0 \end{cases}$

Explanation
The element $0 \in \Dom h$ is defined twice:
 * $h \paren 0 = 0 + 1 = 1$

and:
 * $h \paren 0 = 0$