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John Cook is an applied mathematician working in Houston, Texas. His career has been a blend of research, software development, consulting, and management. John is a DZone MVB and is not an employee of DZone and has posted 171 posts at DZone. You can read more from them at their website. View Full User Profile

Mutually Odd Functions

05.13.2013
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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?

Published at DZone with permission of John Cook, author and DZone MVB. (source)

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