Question

There is exactly one function that is both even and odd. Can anyone tell me what function that is?

Answer 1

undoubtedly,the only function which is both even and odd is constant function i.e. f(x)=0 for all value of x.

Answer 2

Yes, it is f(x)=0, because f(x)=-f(-x)=-0 (Odd) and f(x)=f(-x)=0 (Even)

Answer 3

The proof is pretty simple: since f(x) should be even you have by hypothesis: f(x)=f(-x) [1]. Since it should also be odd you also have: f(x)=-f(-x) [2]. That implies: f(-x)=-f(-x) or 2 f(-x) = 0, that is f(-x)=0. Now if you call z=-x you recognize that f(z)=0, but that means that f does not really depend on z and thus it doesn't depend on -x too, because z=-x. Therefore f(-x)=f(x)=0. It would have been much simpler to sum [1] and [2] and conclude that 2 f(x)=f(-x)-f(-x)=0, that is f(x)=0, directly.