Hi! I have a problem. Is f(x)=0 an even function, or an odd function, or is it possible to be both?

I've been learning about functions in my pre calc class, and we've learned that there are even, odd, or neither functions. Even functions have a symmetry about the Y axis (f(-x)=f(x)), and odd functions have symmetry about the origin (f(-x)=-f(x)). The function f(x)=0 has both. It's graph is simply a horizontal line on the X axis, and is both symmetrical about the Y axis, and also about the origin.

Also, if one were to do the algebra, f(-x) would equal 0, which is the same as f(x), but could also be considered the opposite of f(x), as -1 x 0 = 0.

So, I have come to the conclusion that it is both an even and odd function... but I'm not sure that it's possible for a function to be both even and odd. And would it be called "both even and odd function", or something different? Any comments or suggestions are greatly welcomed =) Whee!

P.S. 0 is a cool number. It's so happy and does lots of things, and kills calculators a lot! Yay!

Zero is even since it can be expressed in the form 2n where n is an integer, zero in this case.

Yes, 0 is even, but is the function f(x)=0 even? It shares properties, and is a valid function, for both even and odd function characteristics. I was wondering if there is such thing as a both even or odd function, because the function f(x)=0 is both... though I don't understand how it can be both =P

Many functions are neither even nor odd. f(x)=0 is both even and odd.

Its the only one that is both odd and even. ( In the real number domain)

From the rules for odd and even

f(-x)=f(x)
and
f(-x)=-f(x)

Equating the right hand side means
f(x) = -f(x)

let N stand for f(x) then
N = -N

which is true only when N = 0 for the real number domain - in modular arrithemetic for example numbers other than 0 work)

and substituting back gives

f(x) = 0

Q.E.D