View source for Decreasing function

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305001.png" /> defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305002.png" /> of real numbers such that the condition

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implies

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Sometimes such a function is called strictly decreasing and the term  "decreasing function"  is applied to functions satisfying for the indicated values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305005.png" /> only the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305006.png" /> (a non-increasing function). Every strictly decreasing function has an inverse function, which is again strictly decreasing. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305007.png" /> is a left-hand (respectively, right-hand) limit point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d0305009.png" /> is non-increasing and if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050010.png" /> is bounded from above (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050011.png" /> is bounded from below), then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050012.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050013.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050015.png" /> has a finite limit; if the given set is not bounded from above (respectively, from below), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050016.png" /> has an infinite limit, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050017.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050018.png" />).



====Comments====
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030500/d03050020.png" /> is decreasing is called increasing (cf. [[Increasing function|Increasing function]]).