Evenness

A function f is called "even" if it holds that:

f[x]=f[-x]

The integral over the range (-a,a) of an even function f is:

∫\limits_{-a}^{a} f[x]⋅dx=2⋅∫\limits_{0}^{a} f[x]⋅dx

Oddness

A function f is called "odd" if it holds that:

f[-x]=-f[x]

The integral over the range (-a,a) of an odd function f is 0:

∫\limits_{-a}^{a} f[x]⋅dx=0

Even and odd

For a function f to be both even and odd, it has to hold that:

f[-x]=-f[x]
f[-x]=f[x]

Which means:

f[x]=-f[x]
2⋅f[x]=0
f[x]=0
The constant function returning zero is the only function that is both even and odd.

Neither

It is possible for a function to be neither even nor odd, but:

Splitting functions

Any function f can be split into one even part and one odd part:

f[x]=÷{f[x]+f[-x]}{2}+÷{f[x]-f[-x]}{2}

The first outer term is even, the second outer term is odd.