Both even function and odd function

Question

I know that 0 is an even function and an odd function. How can I prove f is both even and odd if and only if it is the constant 0 function

Answer 1

Well if $f$ is even then $f(-x) = f(x)$. And if $f$ is odd then $f(-x)= -f(x)$. And if $f$ is both even and odd then $f(x) = f(-x) =-f(x)$.

So....?

So $f(x) = -f(x)$ so $2f(x) = 0$ and $f(x) = 0$.