Functions that are both Odd and Even

Zachary M. Pisano

** Loyola University Maryland
**

The universally accepted statement regarding functions which are both even and odd is:

“The only function which is both even and odd is the constant function which is identically zero (i.e., $f(x)=0$ for all $x$).”

This statement is believed to be correct on the basis that $f(x)=0$ satisfies the definition of both evenness and oddness, namely that both $f(x)=f(-x)$ and $f(-x)=-f(x)$ are true for every $x$ since $f(x)=0$. However, the most common proof of this statement assumes that the domain of $f(x)$ is all real numbers. If the domain of $f$ were to be altered in such a way that any restriction placed on the domain were to have a corresponding restriction equidistant from $x=0$, then the resulting function would be both even and odd.

The simplest restriction that can be placed on the domain of such a function is to exclude $x=0$. This is easily seen in the function $\displaystyle f(x)=\frac{0}{x}$. While this function can be simplified to the form $g(x)=0$ for $x\ne 0$, the two functions are not equal since they have different domains, namely $g(x)=0$ has domain $(-\infty,\infty)$, while $\displaystyle f(x)=\frac{0}{x}$ has domain $(-\infty,0)\cup(0,\infty)$. The graph of $f(x)$ has a removable continuity (i.e., a domain restriction) at the point $(0,0)$.

As a result, the universally accepted statement should be amended to read:

“The only function * whose domain is all real numbers *, which is both even and odd is the constant function which is identically zero (i.e., $f(x)=0$ for all $x$).”

Now that the language has been revised to accommodate for a variety of domains, the way has been paved for an infinite family of functions that are both even and odd.

First, a funtion with a finite number of domain restrictions.

Second, a function with an infinite number of domain restrictions.

For completeness, we also include a function which is identically zero on its domain, but is neither even nor odd.