Monotone function - Encyclopedia of Mathematics

A function of one variable, defined on a subset of the real numbers, whose increment Δf(x)=f(x′)−f(x), for Δx=x′−x>0, does not change sign ...   Login www.springer.com TheEuropeanMathematicalSociety Navigation Mainpage PagesA-Z StatProbCollection Recentchanges Currentevents Randompage Help Projecttalk Requestaccount Tools Whatlinkshere Relatedchanges Specialpages Printableversion Permanentlink Pageinformation Namespaces Page Discussion Variants Views View Viewsource History Actions Monotonefunction FromEncyclopediaofMathematics Jumpto:navigation, search Afunctionofonevariable,definedonasubsetoftherealnumbers,whoseincrement$\Deltaf(x)=f(x^\prime)-f(x)$, for$\Deltax=x^\prime-x>0$, doesnotchangesign,thatis,iseitheralwaysnegativeoralwayspositive.If$\Deltaf(x)$ isstrictlygreater(less)thanzerowhen$\Deltax>0$, thenthefunctioniscalledstrictlymonotone(seeIncreasingfunction;Decreasingfunction).Thevarioustypesofmonotonefunctionsarerepresentedinthefollowingtable.

$\Deltaf(x)\geq0$ Increasing(non-decreasing) $\Deltaf(x)\leq0$ Decreasing(non-increasing) $\Deltaf(x)>0$ Strictlyincreasing $\Deltaf(x)<0$ Strictlydecreasing Ifateachpointofaninterval$f$ hasaderivativethatdoesnotchangesign(respectively,isofconstantsign),then$f$ ismonotone(strictlymonotone)onthisinterval. Theideaofamonotonefunctioncanbegeneralizedtofunctionsofvariousclasses.Forexample,afunction$f(x_{1}\dotsx_{n})$ definedon$\mathbfR^{n}$ iscalledmonotoneifthecondition$x_{1}\leqx_{1}^\prime\dotsx_{n}\leqx_{n}^\prime$ impliesthateverywhereeither$f(x_{1}\dotsx_{n})\leqf(x_{1}^\prime\dotsx_{n}^\prime)$ or$f(x_{1}\dotsx_{n})\geqf(x_{1}^\prime\dotsx_{n}^\prime)$ everywhere.Amonotonefunctioninthealgebraoflogicisdefinedsimilarly. Amonotonefunctionofmanyvariables,increasingordecreasingatsomepoint,isdefinedasfollows.Let$f$ bedefinedonthe$n$- dimensionalclosedcube$Q^{n}$, let$x_{0}\inQ^{n}$ andlet$E_{t}=\{{x}:{f(x)=t,x\inQ^{n}}\}$ bealevelsetof$f$. Thefunction$f$ iscalledincreasing(respectively,decreasing)at$x_{0}$ ifforany$t$ andany$x^\prime\inQ^{n}\setminusE_{t}$ notseparatedin$Q^{n}$ by$E_{t}$ from$x_{0}$, therelation$f(x^\prime)t$) holds,andforany$x^{\prime\prime}\inQ^{n}\setminusE_{t}$ thatisseparatedin$Q^{n}$ by$E_{t}$ from$x_{0}$, therelation$f(x^{\prime\prime})>t$( respectively,$f(x^{\prime\prime})