Show that the only function which is both even and odd is [itex]f(x)=0[/itex]
2. The attempt at a solution
Since [itex]f(x)=0[/itex] is [itex]f(x)=0x[/itex] 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 [itex]f(x)=Ax[/itex] [itex]A\neq0[/itex] 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 [itex]A\neq0[/itex] 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)?