The floor of a real number *x* is the largest integer *n ≤ x*, written ⌊x⌋.

The ceiling of a real number *x* is the smallest integer *n ≥ x*, written ⌈x⌉.

The floor and ceiling have the following symmetric relationship:

⌊-*x*⌋ = -⌈*x*⌉

⌈-*x*⌉ = -⌊*x*⌋

The floor and ceiling functions are not odd, but as a pair they satisfy a generalized parity condition:

*f*(-*x*) = -*g*(*x*)

*g*(-*x*) = -*f*(*x*)

If the functions f and g are equal, then each is an odd function. But in general f and g could be different, as with floor and ceiling.

Is there an established name for this sort of relation? I thought of “mutually odd” because it reminds me of mutual recursion.

Can you think of other examples of mutually odd functions?

**Related posts**:

Saved by symmetry

Odd numbers in odd bases

The power of parity

Am I seriously misunderstanding something, or does _every_ function have a function with which it’s mutually odd? In general any function on the reals f(x) can be written as o(x)+e(x) where o(x) is an odd function and e(x) is an even function. But then if g(x) = o(x)–e(x), we have f(–x)=o(–x)+e(–x)=–o(x)+e(x)=–g(x), and likewise g(–x)=–f(x).

Good observation. I didn’t realize that it’s that simple.

I feel like there should be a category way of thinking about this.

It’s even a bit easier than that. The mop (mutually odd partner) of f is g defined by

g(x) = -f(-x)

In fact g(x) = -f(-x).

Btw, these two operations have an interesting commutator

[⌊, ⌈]x = [⌈, ⌊]x = 1

Amplifying what John Moeller pointed out: for any real function on a symmetrically defined domain g(x)=-f(-x) defines such a pair, and similarly g(x)=+f(-x) defines a “mutually even pair”.

one thing that remind me of this mutually-relation is differential over trigonometry function:

d/dx sin(x) = cos(x)

d/dx cos(x) = -sin(x)

d/dx -sin(x) = -cos(x)

d/dx -cos(x) = sin(x)

f(-x) = -g(x) implies g(-x) = -f(x):

f(-x) = -g (x);

y = -x;

=> f(y) = -g(-y);

-f(y) = g(-y);