Q:

Is $f(x)=0$ the only function that is both even and odd?

Is $f(x)=0$ the only function that is both even and odd?
Thanks to you.

A:

Let $f$ be an even and an odd function.  That is, $f(-x) = f(x) = -f(x)$.  In particular, from this we have:
$$f(x) = -f(x)$$
Can you go from here?

