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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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