5.2: Funciones monótnicas
- Page ID
- 108786
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Supongamos\(D \subset \mathbb{R}, f: D \rightarrow \mathbb{R},\) y\((a, b) \subset D .\) Nosotros decimos\(f\) está aumentando\((a, b)\) si\(f(x)<f(y)\) cada vez que\(a<x<y<b ;\) decimos\(f\) está disminuyendo en\((a, b)\) si\(f(x)>f(y)\) cada vez que\(a<x<y<b ;\) decimos\(f\) es no decreciente en\((a, b)\) si\(f(x) \leq f(y)\) siempre\(a<x<y<b ;\) y decimos \(f\)no está aumentando\((a, b)\) si\(f(x) \geq f(y)\) cada vez\(a<x<y<b .\) Vamos a decir que\(f\) es monótona en\((a, b)\) si\(f\) es no decreciente o no creciente\((a, b)\) y diremos que\(f\) es estrictamente monótona en\((a, b)\) si\(f\) está aumentando o disminuyendo en\((a, b)\).
Si\(f\) es monótono en\((a, b),\) entonces\(f(c+)\) y\(f(c-)\) existir para cada uno\(c \in(a, b)\).
- Prueba
-
Supongamos\(f\) que no disminuye en\((a, b) .\) Let\(c \in(a, b)\) y let
\[\lambda=\sup \{f(x): a<x<c\}.\]
Tenga en cuenta que\(\lambda \leq f(c)<+\infty .\) Dada alguna debe\(\epsilon>0,\) existir\(\delta>0\) tal que
\[\lambda-\epsilon<f(c-\delta) \leq \lambda .\]
Dado que no\(f\) es decreciente, se deduce que
\[|f(x)-\lambda|<\epsilon\]
siempre que\(x \in(c-\delta, c) .\) Así\(f(c-)=\lambda .\) Un argumento similar muestra que\(f(c+)=\kappa\) donde
\[\kappa=\inf \{f(x): c<x<b\}.\]
Si no\(f\) es creciente, los argumentos similares rinden
\[f(c-)=\inf \{f(x): a<x<c\}\]
y
\[f(c+)=\sup \{f(x): c<x<b\}.\]
Si\(f\) es no decreciente\((a, b)\) y\(a<x<y<b,\) luego
\[f(x+) \leq f(y-).\]
- Prueba
-
Por la proposición anterior,
\[f(x+)=\inf \{f(t): x<t<b\}\]
y
\[f(y-)=\sup \{f(t): a<t<y\}.\]
Dado que no\(f\) es decreciente,
\[\inf \{f(t): x<t<b\}=\inf \{f(t): x<t<y\}\]
y
\[\sup \{f(t): a<t<y\}=\sup \{f(t): x<t<y\}.\]
Así
\[f(x+)=\inf \{f(t): x<t<y\} \leq \sup \{f(t): x<t<y\}=f(y-).\]
Q.E.D.
\(\varphi: \mathbb{Q} \cap[0,1] \rightarrow \mathbb{Z}^{+}\)Sea una correspondencia uno a uno. Definir\(f:[0,1] \rightarrow \mathbb{R}\) por
\[f(x)=\sum_{q \in \mathbb{Q} \cap[0,1]_{q \leq x}} \frac{1}{2^{\varphi(q)}}.\]
a. Demostrar que\(f\) va en aumento\((0,1)\).
b. Demostrar que para cualquier\(x \in \mathbb{Q} \cap(0,1), f(x-)<f(x)\) y\(f(x+)=f(x)\).
c. Demostrarlo por cualquier irracional\(a, 0<a<1, \lim _{x \rightarrow a} f(x)=f(a)\).