# Even and odd function question

1. Feb 1, 2012

### mindauggas

1. The problem statement, all variables and given/known data

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

2. The attempt at a solution

Since $f(x)=0$ is $f(x)=0x$ it is not hard to show that it is odd and even. In order to complete the proof I need to show that this is the only funcion. I know intuitively that if in $f(x)=Ax$ $A\neq0$ then the function is always either odd, even or neither. How should I complete the proof? What should I write?

Maybe: "Since, it is obvious that if $A\neq0$ the function is either ... " But is it really that obvious? Should I use proof by exhaustion and show that in all posible cases this is true (when A is positive, negative, fraction)?

Last edited: Feb 1, 2012
2. Feb 1, 2012

### A. Bahat

If a function f is both odd and even, then f(-x)=f(x) and f(-x)=-f(x), so...

3. Feb 1, 2012

### tiny-tim

hi mindauggas!

start "suppose f(x) is not 0 for all x, then …"