Mutually Odd Functions
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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