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

Does this answer your question? Can there be a function that's even and odd at the same time? — scoopfaze, Feb 03 '20 at 05:47
Even means $f(-x) = f(x)$ and odd means $f(-x) = -f(x)$ so both would mean $-f(x) = f(-x) =f(x)$ and $-f(x) = f(x)$ which means .... what? — fleablood, Feb 03 '20 at 05:51

score 1 · Answer 1 · answered Feb 03 '20 at 05:50

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 $f(x) = 0$.

answered Feb 03 '20 at 05:50

fleablood

1 Answers1