Is there a function such that it is both even and odd?
Is it f(x)=0? any others?
A function is odd if $\displaystyle \forall x, \ f(-x) = -f(x)$
A function is even if $\displaystyle \forall x, \ f(-x) = f(x)$
Assume we have a function g that is both even and odd, then:
$\displaystyle g(-x) = g(x) = -g(-x) \Rightarrow g(-x) = -g(-x) \Rightarrow 2g(-x) = 0$
$\displaystyle \Rightarrow g(x) \equiv 0$