Odd Function Times Even Function is Odd

Theorem
Let $X \subset \R$ be a symmetric set of real numbers:
 * $\forall x \in X: -x \in X$

Let $f: X \to \R$ be an odd function.

Let $g: X \to \R$ be an even function.

Let $f \cdot g$ denote the pointwise product of $f$ and $g$.

Then $\paren {f \cdot g}: X \to \R$ is an odd function.

Proof
Thus, by definition, $\paren {f \cdot g}$ is an odd function.

Also see

 * Even Function Times Even Function is Even
 * Odd Function Times Odd Function is Even