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So I was wondering, is there a name for a function whose output is always less than or equal to its input ($f(x)≤x$)? I know there is a name for functions that satisfy $x_1<x_2\rightarrow f(x_1)<f(x_2)$ (monotonic) so I figured there would be a name for $f(x)≤x$. Does anyone know what it is? A good example of this would be the greatest integer function $\lfloor x\rfloor$. Thank you.

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A function satisfying "$x_1<x_2\rightarrow f(x_1)<f(x_2)$" is called strictly increasing, not monotonic. –  Chris Eagle Dec 13 '11 at 15:58
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I've seen functions satisfying "$x \le f(x)$" called inflationary, progressive, or extensive. Thus one could use deflationary or regressive for your property, but I haven't actually seen such usage. –  Chris Eagle Dec 13 '11 at 16:04
    
In mathematics, a monotonic function (or monotone function) is a function that preserves the given order: en.wikipedia.org/wiki/Monotonic_function. Oh and thanks for your comment. :) –  Hautdesert Dec 13 '11 at 21:29
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Regressive function appears to be a common term in set theory: e.g. Regressive function on an ordinal and set of infinite cardinals admits an injective regressive function. It requires strict inequality, but if you use it outside of set theory, you are going to explain the term anyway, which will give you an opportunity to say that only $\leq$ is required.

In mathematics, a monotonic function (or monotone function) is a function that preserves the given order: en.wikipedia.org/wiki/Monotonic_function

Wikipedia is great, but it is not a holy script. The function $f(x)=-x$ is monotone, although it reverses the order of $\mathbb R$ instead of preserving it.

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