RA Monotone functions and continuity - Jiří Lebl

If a function is either increasing or decreasing, we say it is monotone. If it is strictly increasing or strictly decreasing, ... 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\newcommand{\unitfrac}[3][\!\!]{#1\,\,{{}^{#2}}\!/\!{{}_{#3}}} \newcommand{\unit}[2][\!\!]{#1\,\,#2} \newcommand{\noalign}[1]{} \newcommand{\qed}{\qquad\Box} \newcommand{\qedhere}{} \newcommand{\widebar}[1]{\overline{#1}} \newcommand{\lt}{} \newcommand{\amp}{&} \) FrontMatterColophon 0Introduction Aboutthisbook Aboutanalysis Basicsettheory 1RealNumbers Basicproperties Thesetofrealnumbers Absolutevalueandboundedfunctions Intervalsandthesizeof\(\R\) Decimalrepresentationofthereals 2SequencesandSeries Sequencesandlimits Factsaboutlimitsofsequences Limitsuperior,limitinferior,andBolzano–Weierstrass Cauchysequences Series Moreonseries 3ContinuousFunctions Limitsoffunctions Continuousfunctions Min-maxandintermediatevaluetheorems Uniformcontinuity Limitsatinfinity Monotonefunctionsandcontinuity 4TheDerivative Thederivative Meanvaluetheorem Taylor'stheorem Inversefunctiontheorem 5TheRiemannIntegral TheRiemannintegral Propertiesoftheintegral Fundamentaltheoremofcalculus Thelogarithmandtheexponential Improperintegrals 6SequencesofFunctions Pointwiseanduniformconvergence Interchangeoflimits Picard'stheorem 7MetricSpaces Metricspaces Openandclosedsets Sequencesandconvergence Completenessandcompactness Continuousfunctions FixedpointtheoremandPicard'stheoremagain 8SeveralVariablesandPartialDerivatives Vectorspaces,linearmappings,andconvexity Analysiswithvectorspaces Thederivative Continuityandthederivative Inverseandimplicitfunctiontheorems Higherorderderivatives 9One-dimensionalIntegralsinSeveralVariables Differentiationundertheintegral Pathintegrals Pathindependence 10MultivariableIntegral Riemannintegraloverrectangles IteratedintegralsandFubinitheorem Outermeasureandnullsets ThesetofRiemannintegrablefunctions Jordanmeasurablesets Green'stheorem Changeofvariables 11FunctionsasLimits Complexnumbers Swappinglimits Powerseriesandanalyticfunctions Thecomplexexponentialandthetrigonometricfunctions Fundamentaltheoremofalgebra EquicontinuityandtheArzelà–Ascolitheorem TheStone–Weierstrasstheorem Fourierseries BackMatter FurtherReading Index CreatedwithPreTeXt Section3.6Monotonefunctionsandcontinuity Note:1lecture(optional,cansafelybeomittedunlessSection 4.4isalsocovered,requiresSection 3.5) Definition3.6.1. Let\(S\subset\R\text{.}\)Wesay\(f\colonS\to\R\)isincreasing(resp.strictlyincreasing)if\(x,y\inS\)with\(xc,x\inS\} \qquad\text{and}\qquad \lim_{x\toc^+}g(x)=\sup\{g(x):x>c,x\inS\}. \end{equation*} If\(\infty\)isaclusterpointof\(S\text{,}\)then \begin{equation*} \lim_{x\to\infty}f(x)=\sup\{f(x):x\inS\} \qquad\text{and}\qquad \lim_{x\to\infty}g(x)=\inf\{g(x):x\inS\}. \end{equation*} If\(-\infty\)isaclusterpointof\(S\text{,}\)then \begin{equation*} \lim_{x\to-\infty}f(x)=\inf\{f(x):x\inS\} \qquad\text{and}\qquad \lim_{x\to-\infty}g(x)=\sup\{g(x):x\inS\}. \end{equation*} Namely,alltheone-sidedlimitsexistwhenevertheymakesense.Formonotonefunctionstherefore,whenwesaytheleft-handlimit\(x\toc^-\)exists,wemeanthat\(c\)isaclusterpointof\(S\cap(-\infty,c)\text{,}\)andsamefortheright-handlimit. Proof. Letusassume\(f\)isincreasing,andwewillshowthefirstequality.Therestoftheproofisverysimilarandisleftasanexercise. Let\(a:=\sup\{f(x):xM\text{.}\)As\(f\)isincreasing,\(f(x)\geqf(x_M)>M\)forall\(x\inS\)with\(x>x_M\text{.}\)Ifwetake\(\delta:=c-x_M>0\text{,}\)thenweobtainthedefinitionofthelimitgoingtoinfinity. Nextsuppose\(a0\)begiven.Because\(a\)isthesupremumand\(S\cap(-\infty,c)\)isnonempty,\(a\in\R\)andthereexistsan\(x_\epsilon\inS\text{,}\)\(x_\epsilona-\epsilon\text{.}\)As\(f\)isincreasing,if\(x\inS\)and\(x_\epsilonc\bigr\}. \end{equation*} As\(c\)isadiscontinuity,\(ac\text{,}\)then\(f(x)\geqb\text{.}\)Thereforenopointin\((a,b)\setminus\{f(c)\}\)isin\(f(I)\text{.}\)Howeverthereexists\(x_1\inI\text{,}\)\(x_1c\text{,}\)so\(f(x_2)\geqb\text{.}\)Both\(f(x_1)\)and\(f(x_2)\)arein\(f(I)\text{,}\)buttherearepointsinbetweenthemthatarenotin\(f(I)\text{.}\)So\(f(I)\)isnotaninterval.SeeFigure 3.8. When\(c\inI\)isanendpoint,theproofissimilarandisleftasanexercise. Figure3.8.Increasingfunction\(f\colonI\to\R\)discontinuityat\(c\text{.}\) Astrikingpropertyofmonotonefunctionsisthattheycannothavetoomanydiscontinuities. Corollary3.6.4. Let\(I\subset\R\)beanintervaland\(f\colonI\to\R\)bemonotone.Then\(f\)hasatmostcountablymanydiscontinuities.Proof. Let\(E\subsetI\)bethesetofalldiscontinuitiesthatarenotendpointsof\(I\text{.}\)Asthereareonlytwoendpoints,itisenoughtoshowthat\(E\)iscountable.Withoutlossofgenerality,suppose\(f\)isincreasing.Wewilldefineaninjection\(h\colonE\to\Q\text{.}\)Foreach\(c\inE\)theone-sidedlimitsof\(f\)bothexistas\(c\)isnotanendpoint.Let \begin{equation*} a:=\lim_{x\toc^-}f(x)=\sup\bigl\{f(x):x\inI,xc\bigr\}. \end{equation*} As\(c\)isadiscontinuity,wehave\(ac\text{,}\)thenthereexistan\(x\inI\)with\(cb\text{.}\)Similarlyif\(df(y)\)if\(f\)isstrictlydecreasing,so\(f(x)\not=f(y)\text{.}\)Hence,\(f\)musthaveaninverse\(f^{-1}\)definedonitsrange. Proposition3.6.6. If\(I\subset\R\)isanintervaland\(f\colonI\to\R\)isstrictlymonotone,thentheinverse\(f^{-1}\colonf(I)\toI\)iscontinuous.Proof. Letussuppose\(f\)isstrictlyincreasing.Theproofisalmostidenticalforastrictlydecreasingfunction.Since\(f\)isstrictlyincreasing,sois\(f^{-1}\text{.}\)Thatis,if\(f(x)< f(y)\text{,}\)thenwemusthave\(xc,y\inf(I)\bigr\} = \inf\bigl\{x\inI:f(x)>c\bigr\}. \end{aligned} \end{equation*} Wehave\(x_0\leqx_1\)as\(f^{-1}\)isincreasing.Forall\(x\inI\)where\(x>x_0\text{,}\)wehave\(f(x)\geqc\text{.}\)As\(f\)isstrictlyincreasing,wemusthave\(f(x)>c\)forall\(x\inI\)where\(x>x_0\text{.}\)Therefore, \begin{equation*} \{x\inI:x>x_0\}\subset\bigl\{x\inI:f(x)>c\bigr\}. \end{equation*} Theinfimumoftheleft-handsetis\(x_0\text{,}\)andtheinfimumoftheright-handsetis\(x_1\text{,}\)soweobtain\(x_0\geqx_1\text{.}\)So\(x_1=x_0\text{,}\)and\(f^{-1}\)iscontinuousat\(c\text{.}\) Ifoneoftheone-sidedlimitsdoesnotexist,theargumentissimilarandisleftasanexercise. Example3.6.7. Thepropositiondoesnotrequire\(f\)itselftobecontinuous.Let\(f\colon\R\to\R\)bedefinedby \begin{equation*} f(x):= \begin{cases} x&\text{if}x<0,\\ x+1&\text{if}x\geq0.\\ \end{cases} \end{equation*} Thefunction\(f\)isnotcontinuousat\(0\text{.}\)Theimageof\(I=\R\)istheset\((-\infty,0)\cup[1,\infty)\text{,}\)notaninterval.Then\(f^{-1}\colon(-\infty,0)\cup[1,\infty) \to\R\)canbewrittenas \begin{equation*} f^{-1}(y)= \begin{cases} y&\text{if}y<0,\\ y-1&\text{if}y\geq1. \end{cases} \end{equation*} Itisnotdifficulttoseethat\(f^{-1}\)isacontinuousfunction.SeeFigure 3.10forthegraphs. Figure3.10.Graphof\(f\)ontheleftand\(f^{-1}\)ontheright.Noticewhathappenswiththepropositionif\(f(I)\)isaninterval.Inthatcase,wecouldsimplyapplyCorollary 3.6.3toboth\(f\)and\(f^{-1}\text{.}\)Thatis,if\(f\colonI\toJ\)isanontostrictlymonotonefunctionand\(I\)and\(J\)areintervals,thenboth\(f\)and\(f^{-1}\)arecontinuous.Furthermore,\(f(I)\)isanintervalpreciselywhen\(f\)iscontinuous. Subsection3.6.3Exercises Exercise3.6.1. Suppose\(f\colon[0,1]\to\R\)ismonotone.Prove\(f\)isbounded. Exercise3.6.2. FinishtheproofofProposition 3.6.2.Hint:Youcanhalveyourworkbynoticingthatif\(g\)isdecreasing,then\(-g\)isincreasing. Exercise3.6.3. FinishtheproofofCorollary 3.6.3. Exercise3.6.4. ProvetheclaimsinExample 3.6.5. Exercise3.6.5. FinishtheproofofProposition 3.6.6. Exercise3.6.6. Suppose\(S\subset\R\text{,}\)and\(f\colonS\to\R\)isanincreasingfunction.Prove: If\(c\)isaclusterpointof\(S\cap(c,\infty)\text{,}\)then\(\lim\limits_{x\toc^+}f(x)0\)suchthat\(f(x)\geqax+b\)forall\(x\inI\)and\(f(c)=ac+b\text{.}\)Showthat\(f\)isstrictlyincreasing. Exercise3.6.8. Suppose\(I\)and\(J\)areintervalsand\(f\colonI\toJ\)isacontinuous,bijective(one-to-oneandonto)function.Showthat\(f\)isstrictlymonotone. Exercise3.6.9. Consideramonotonefunction\(f\colonI\to\R\)onaninterval\(I\text{.}\)Provethatthereexistsafunction\(g\colonI\to\R\)suchthat\(\lim\limits_{x\toc^-}g(x)=g(c)\)forall\(c\)in\(I\)exceptthesmaller(left)endpointof\(I\text{,}\)andsuchthat\(g(x)=f(x)\)forallbutcountablymany\(x\inI\text{.}\) Exercise3.6.10. Let\(S\subset\R\)beasubset.If\(f\colonS\to\R\)isincreasingandbounded,thenshowthatthereexistsanincreasing\(F\colon\R\to\R\)suchthat\(f(x)=F(x)\)forall\(x\inS\text{.}\) Findanexampleofastrictlyincreasingbounded\(f\colonS\to\R\)suchthatanincreasing\(F\)asaboveisneverstrictlyincreasing. Exercise3.6.11. (Challenging)  Findanexampleofanincreasingfunction\(f\colon[0,1]\to\R\)thathasadiscontinuityateachrationalnumber.Thenshowthattheimage\(f([0,1])\)containsnointerval.Hint:Enumeratetherationalnumbersanddefinethefunctionwithaseries. Exercise3.6.12. Suppose\(I\)isanintervaland\(f\colonI\to\R\)ismonotone.Showthat\(\R\setminusf(I)\)isacountableunionofdisjointintervals. Exercise3.6.13. Suppose\(f\colon[0,1]\to(0,1)\)isincreasing.Showthatforevery\(\epsilon>0\text{,}\)thereexistsastrictlyincreasing\(g\colon[0,1]\to(0,1)\)suchthat\(g(0)=f(0)\text{,}\)\(f(x)\leqg(x)\)forall\(x\text{,}\)and\(g(1)-f(1)