# Thread: even and odd functions

1. ## even and odd functions

Is there a function such that it is both even and odd?

Is it f(x)=0? any others?

2. A function is odd if $\forall x, \ f(-x) = -f(x)$

A function is even if $\forall x, \ f(-x) = f(x)$

Assume we have a function g that is both even and odd, then:

$g(-x) = g(x) = -g(-x) \Rightarrow g(-x) = -g(-x) \Rightarrow 2g(-x) = 0$

$\Rightarrow g(x) \equiv 0$