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?

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