Monotonous real function

A monotonically increasing real function (red) and a monotonically decreasing real function (blue)

A monotonous real function is a real-valued function of a real variable in which the function value either always increases or always decreases when the argument is increased. If the function value always increases when the argument is increased, the function is called strictly monotonically increasing ; if the function value always increases or remains the same, it is called monotonically increasing . Similarly, a function is called strictly monotonically decreasing if its function value always falls when the argument is increased, and monotonically decreasing if it always falls or remains the same. Real monotonic functions are classic examples of monotonic mappings . ${\displaystyle f(x)}$ ${\displaystyle x}$

definition

A function where is a subset of is called ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle D}$ ${\displaystyle \mathbb {R} }$

• monotonically increasing if it applies to all with that . ${\displaystyle x,y\in D}$ ${\displaystyle x\leq y}$ ${\displaystyle f(x)\leq f(y)}$
• strictly increasing if for all to apply that . ${\displaystyle x,y\in D}$ ${\displaystyle x<y}$ ${\displaystyle f(x)<f(y)}$
• monotonically decreasing , if it applies to all with that . ${\displaystyle x,y\in D}$ ${\displaystyle x\leq y}$ ${\displaystyle f(x)\geq f(y)}$
• strictly decreasing if for all to apply that . ${\displaystyle x,y\in D}$ ${\displaystyle x<y}$ ${\displaystyle f(x)>f(y)}$
• monotonous if it either increases monotonically or decreases monotonically.
• strictly monotonous if it either increases strictly monotonically or falls strictly monotonously.

Sometimes the not strict monotony terms are also defined for. The two definitions are equivalent. A synonym for “strict” is also found “strictly”, monotonically falling is sometimes also called antitone , just as monotonically increasing is also called isotonic . There is also the term “growing” instead of “rising”. ${\displaystyle x<y}$

Examples

Graph of the function ${\displaystyle f(x)=x^{2}}$

Graph of the function ${\displaystyle f(x)=\ln(x)}$

• The function is strictly decreasing to monotonous. Namely , so is and . The condition that should be is equivalent to . But it is with the third binomial formula ${\displaystyle f(x)=x^{2}}$ ${\displaystyle (-\infty ,0]}$ ${\displaystyle x<y\leq 0}$ ${\displaystyle x-y<0}$ ${\displaystyle x+y<0}$ ${\displaystyle f(x)=x^{2}>f(y)=y^{2}}$ ${\displaystyle x^{2}-y^{2}>0}$

${\displaystyle x^{2}-y^{2}=\underbrace {(x+y)} _{<0}\underbrace {(x-y)} _{<0}>0}$,

so is strictly monotonous . The proof that is strictly increasing monotonically works analogously, but with the argument that if is. This means that the function is not monotonic because it does not have a fixed monotonic behavior on this interval. ${\displaystyle f}$ ${\displaystyle (-\infty ,0]}$ ${\displaystyle f}$ ${\displaystyle [0,\infty )}$ ${\displaystyle x+y>0}$ ${\displaystyle y>x\geq 0}$ ${\displaystyle [-1,1]}$

• The logarithm is strictly increasing monotonically . is again equivalent to . Then ${\displaystyle (0,\infty )}$ ${\displaystyle \ln(x)<\ln(y)}$ ${\displaystyle \ln(x)-\ln(y)<0}$

${\displaystyle \ln(x)-\ln(y)=\ln(x/y)<0}$,

if there is then and accordingly . So is . Thus the logarithm is strictly monotonically increasing and therefore also strictly monotonously. ${\displaystyle x<y}$ ${\displaystyle 0<{\tfrac {x}{y}}<1}$ ${\displaystyle \ln(x/y)<0}$ ${\displaystyle \ln(x)<\ln(y)}$

• The function

${\displaystyle f(x)={\begin{cases}x^{2}&{\text{ für }}x<0\\0&{\text{ für }}x\geq 0\end{cases}}}$

is monotonically decreasing on the interval , but not strictly monotonically decreasing. The detection of monotony in the left half of the interval follows the first example, on the interval , however, and thus may apply no strict monotonicity. Thus the function is monotonically decreasing and therefore also monotonous. ${\displaystyle [-1,1]}$ ${\displaystyle [0,1]}$ ${\displaystyle f(x)-f(y)=0}$

properties

For a real monotonic function with : ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle D\subseteq \mathbb {R} }$

• Strictly monotonous functions are always injective , so they only take each value once at most. If strictly monotonic and an interval and the set of images is bijective . Therefore the inverse function always exists for strictly monotonic functions . For example, the sine function increases strictly monotonically on the interval . If one restricts the amount of images to the interval , it is bijective and thus invertible. The inverse function is then the arcsine . ${\displaystyle f\colon I\to \mathbb {R} }$ ${\displaystyle I}$ ${\displaystyle I':=f(I)}$ ${\displaystyle f\colon I\to I'}$ ${\displaystyle [-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}]}$ ${\displaystyle [-1,1]}$ ${\displaystyle \arcsin :[-1,1]\to [-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}]}$
• It has a left-hand and right-hand limit value in each accumulation point of its definition range . ${\displaystyle D}$
• If a strictly monotonic function converges with , then its function value is different in its entire domain of . ${\displaystyle \textstyle \lim _{x\to \pm \infty }f(x)=L}$ ${\displaystyle L}$

(indirect) proof

A. Prerequisite: ${\displaystyle \textstyle \lim _{x\to -\infty }f(x)=L.\quad }$ Assumption : There is a with . ${\displaystyle x_{0}\in D}$ ${\displaystyle f(x_{0})=L}$ If it is strictly monotonically increasing, there exists with such that ; ${\displaystyle f}$ ${\displaystyle D\ni x_{1}<x_{0}}$ ${\displaystyle f(x_{1})<f(x_{0})=L,}$ ${\displaystyle L-f(x_{1})=:\epsilon >0}$ next is for everyone ${\displaystyle f(x)<f(x_{1})=L-\epsilon }$ ${\displaystyle D\ni x<x_{1};\quad }$ is strictly monotonically decreasing, there exists with such that ; ${\displaystyle f}$ ${\displaystyle D\ni x_{1}<x_{0}}$ ${\displaystyle f(x_{1})>f(x_{0})=L,}$ ${\displaystyle f(x_{1})-L=:\epsilon >0}$ next is for everyone ${\displaystyle f(x)>f(x_{1})=L+\epsilon }$ ${\displaystyle D\ni x<x_{1}.\quad }$ Both considerations can be summarized in a formulation that also allows the following: ${\displaystyle f(x)=L\pm \epsilon }$ Because of the strict monotony of there exists with such ; ${\displaystyle f}$ ${\displaystyle D\ni x_{1}<x_{0}}$ ${\displaystyle f(x_{1})\neq f(x_{0})=L,}$ ${\displaystyle |f(x_{1})-L|=:\epsilon >0}$ next is for everyone (1) ${\displaystyle f(x)\notin U_{\epsilon }(L)}$ ${\displaystyle D\ni x<x_{1}.\quad }$ Because of the convergence of there exists such that for all (2) ${\displaystyle f}$ ${\displaystyle S}$ ${\displaystyle f(x)\in U_{\epsilon }(L)}$ ${\displaystyle D\ni x<S.\quad }$ With (1) and (2) we have for all both also (contradiction), q. e. d. ${\displaystyle x<\min(x_{1},S)}$ ${\displaystyle f(x)\notin U_{\epsilon }(L)}$ ${\displaystyle f(x)\in U_{\epsilon }(L)}$ B. Requirement: ${\displaystyle \textstyle \lim _{x\to +\infty }f(x)=L.\quad }$ Assumption : There is a with . ${\displaystyle x_{0}\in D}$ ${\displaystyle f(x_{0})=L}$ If it is strictly monotonically increasing, there exists with such that ; ${\displaystyle f}$ ${\displaystyle D\ni x_{1}>x_{0}}$ ${\displaystyle f(x_{1})>f(x_{0})=L,}$ ${\displaystyle f(x_{1})-L=:\epsilon >0}$ next is for everyone ${\displaystyle f(x)>f(x_{1})=L+\epsilon }$ ${\displaystyle D\ni x>x_{1};\quad }$ is strictly monotonically decreasing, there exists with such that ; ${\displaystyle f}$ ${\displaystyle D\ni x_{1}>x_{0}}$ ${\displaystyle f(x_{1})<f(x_{0})=L,}$ ${\displaystyle L-f(x_{1})=:\epsilon >0}$ next is for everyone ${\displaystyle f(x)<f(x_{1})=L-\epsilon }$ ${\displaystyle D\ni x>x_{1}.\quad }$ Both considerations can be summarized in a formulation that also allows the following: ${\displaystyle f(x)=f(x_{0})=L\pm \epsilon }$ Because of the strict monotony of there exists with such ; ${\displaystyle f}$ ${\displaystyle D\ni x_{1}>x_{0}}$ ${\displaystyle f(x_{1})\neq L,}$ ${\displaystyle |f(x_{1})-L|=:\epsilon >0}$ next is for everyone (1 ') ${\displaystyle f(x)\notin U_{\epsilon }(L)}$ ${\displaystyle D\ni x>x_{1}.\quad }$ Because of the convergence of there exists such that for all (2 ') ${\displaystyle f}$ ${\displaystyle S}$ ${\displaystyle f(x)\in U_{\epsilon }(L)}$ ${\displaystyle D\ni x>S.\quad }$ With (1 ') and (2') applies to all both including (opposition) q. e. d. ${\displaystyle x>\max(x_{1},S)}$ ${\displaystyle f(x)\notin U_{\epsilon }(L)}$ ${\displaystyle f(x)\in U_{\epsilon }(L)}$

• It can only have jump points as discontinuities.
• The set of jump points in their domain can be counted , but does not necessarily have to be finite.
• It is differentiable almost everywhere , i. H. the set of places at which is not differentiable forms a Lebesgue zero set . ${\displaystyle f}$
• A monotonic function defined in the interval can be Riemann integrated there . ${\displaystyle D=[a,b]}$
• For every monotonically increasing function, we have for any . This property is partly used to generalize the monotony, see last section. ${\displaystyle (x-y)(f(x)-f(y))\geq 0}$ ${\displaystyle x,y\in D}$
• The monotony of real functions is a special case of monotonic mapping . In the case of a monotonically decreasing function, the two ordered sets are then and , the mapping is the function . ${\displaystyle (\mathbb {R} ,\leq )}$ ${\displaystyle (\mathbb {R} ,\geq )}$ ${\displaystyle f}$

Derivatives as a criterion for monotony

criteria

If the function is differentiable, the derivative can be used as a monotony criterion. The criteria for strict monotony are: ${\displaystyle f\colon (a,b)\to \mathbb {R} }$

• Is for everyone , it grows in strictly monotonous. ${\displaystyle f'(x)>0}$ ${\displaystyle x\in (a,b)}$ ${\displaystyle f}$ ${\displaystyle (a,b)}$
• Is for everyone , it falls in strictly monotonous. ${\displaystyle f'(x)<0}$ ${\displaystyle x\in (a,b)}$ ${\displaystyle f}$ ${\displaystyle (a,b)}$

It should be noted that this criterion is sufficient, but not necessary. There are also strictly monotone functions whose derivative becomes zero, an example is given below. A tightening of these criteria can be formulated with additional requirements:

• It is ( ) for all and the derivative is not constantly zero on any real subinterval (where a real interval is an interval with more than one element) if and only if is strictly monotonically increasing (strictly monotonically decreasing). ${\displaystyle f'(x)\geq 0}$ ${\displaystyle f'(x)\leq 0}$ ${\displaystyle x\in (a,b)}$ ${\displaystyle f}$

The criteria for monotony are:

• ${\displaystyle f'(x)\geq 0}$for all if and only if in increases monotonically. ${\displaystyle x\in (a,b)}$ ${\displaystyle f}$ ${\displaystyle (a,b)}$
• ${\displaystyle f'(x)\leq 0}$for all if and only if in falls monotonously. ${\displaystyle x\in (a,b)}$ ${\displaystyle f}$ ${\displaystyle (a,b)}$

These criteria are equivalences.

All of the criteria mentioned can be expanded further: If continuously on (or or ), then the statement about monotony also applies to the interval (or or ). ${\displaystyle f}$ ${\displaystyle [a,b)}$ ${\displaystyle (a,b]}$ ${\displaystyle [a,b])}$ ${\displaystyle [a,b)}$ ${\displaystyle (a,b]}$ ${\displaystyle [a,b])}$

Examples

The graph of the function . The function grows strictly monotonically. ${\displaystyle f(x)=x^{3}}$

• For the exponential function is for everyone . So it is strictly increasing monotonously. ${\displaystyle f(x)=e^{x}}$ ${\displaystyle f'(x)=e^{x}>0}$ ${\displaystyle x\in \mathbb {R} }$
• The function has the derivative , it becomes zero. But the function is strictly increasing monotonically. For if and if they have the same sign, then is ${\displaystyle f(x)=x^{3}}$ ${\displaystyle f'(x)=3x^{2}}$ ${\displaystyle x=0}$ ${\displaystyle x<y}$ ${\displaystyle x,y}$

${\displaystyle x^{3}-y^{3}=(x-y)(x^{2}+y^{2}+xy)<0}$.

If both have different signs, then is direct . This is an example of how the first two criteria are sufficient but not necessary. The third criterion applies here: the derivative of the function only disappears in the point and is otherwise greater than or equal to zero. This is equivalent to the monotonous growth of . ${\displaystyle x^{3}-y^{3}<0}$ ${\displaystyle x_{0}=0}$ ${\displaystyle f}$

Inverse function

Let be an interval and be strictly monotonically increasing / decreasing and continuous. Then: ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle f\colon I\to \mathbb {R} }$

• the amount of images an interval, ${\displaystyle I':=f\left(I\right)}$
• ${\displaystyle f\colon I\rightarrow I'}$ bijective,
• the inverse function strictly monotonically increasing / decreasing and continuous, ${\displaystyle f^{-1}\colon I'\rightarrow I}$
• ${\displaystyle f^{-1}\left(a\right)<b\iff a<f\left(b\right)}$when growing and
• ${\displaystyle f^{-1}\left(a\right)<b\iff a>f\left(b\right)}$when falling.

Generalizations

K-monotonic functions

→ Main article : K-monotonic function

If one generalizes the concept of monotony for functions , then one defines on the one real cone and considers the generalized inequality defined by it and the strictly generalized inequality as well as a convex set . Then a function is called ${\displaystyle h\colon \mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle K}$ ${\displaystyle \preccurlyeq _{K}}$ ${\displaystyle x\prec _{K}y}$ ${\displaystyle D}$ ${\displaystyle h\colon \mathbb {R} ^{n}\supset D\to \mathbb {R} }$

• K-monotonically increasing ( K-monotonically decreasing ) if it applies to all with that (or ) ${\displaystyle x,y\in D}$ ${\displaystyle x\preccurlyeq _{K}y}$ ${\displaystyle f(x)\leq f(y)}$ ${\displaystyle f(x)\geq f(y)}$
• strictly K-monotonically increasing ( strictly K-monotonically decreasing ) if it holds for all that (or ) is. ${\displaystyle x\prec _{K}y}$ ${\displaystyle f(x)<f(y)}$ ${\displaystyle f(x)>f(y)}$

If one chooses the (the space of all real symmetrical matrices) as the vector space and the semidefinite cone as the cone (or the Loewner partial order as the generalized inequality ), one obtains the matrix-monotonic functions . ${\displaystyle S^{n}}$

Monotonic functions between vector spaces of the same dimension

One possibility to generalize monotony for functions is to demand that if for is that then for a monotonically increasing function that is should apply . The formulation of monotonically decreasing functions and the strict versions follows analogously. This procedure corresponds to the generalization of the order on to the component-wise partial order on . ${\displaystyle h\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle x=(x_{1},\dots ,x_{n})^{T},\,y=(y_{1},\dots ,y_{n})^{T}}$ ${\displaystyle x_{i}\leq y_{i}}$ ${\displaystyle i=1,\dots ,n}$ ${\displaystyle h_{i}(x)\leq h_{i}(y)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{n}}$

Alternatively, one can generalize the property of monotonically increasing real functions that holds for any that is. This then leads to the following concept of monotony: given and a function . The function is called ${\displaystyle x,y}$ ${\displaystyle (x-y)(f(x)-f(y))\geq 0}$ ${\displaystyle D\subset \mathbb {R} ^{n}}$ ${\displaystyle f\colon D\to \mathbb {R} ^{n}}$

• Monotonously on , when all is considered. ${\displaystyle D}$ ${\displaystyle (x-y)^{T}(f(x)-f(y))\geq 0}$ ${\displaystyle x,y\in D}$
• Strictly monotonous on when applies to everyone . ${\displaystyle D}$ ${\displaystyle (x-y)^{T}(f(x)-f(y))>0}$ ${\displaystyle x,y\in D}$
• Uniformly monotone on when for all with valid. ${\displaystyle D}$ ${\displaystyle (x-y)^{T}(f(x)-f(y))\geq \mu \Vert x-y\Vert ^{2}}$ ${\displaystyle x,y\in D}$ ${\displaystyle x\neq y}$

If one generalizes this further, one obtains the concept of a monotonic operator .

Square monotonic function

The monotony for functions can also be defined using the difference operator . A function is then called a rectangular monotonic function if ${\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle \Delta _{a}^{b}}$

${\displaystyle a\leq b\implies \Delta _{a}^{b}F\geq 0}$

applies.

literature

• Otto Forster: Analysis 1 . Differential and integral calculus of a variable. 11th, expanded edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-00316-6 , doi : 10.1007 / 978-3-658-00317-3 .
• Konrad Königsberger: Analysis 1 . 6th, revised edition. Springer-Verlag, Berlin Heidelberg New York 2004, ISBN 3-540-40371-X .
• Stephen Boyd, Lieven Vandenberghe: Convex Optimization . Cambridge University Press, Cambridge, New York, Melbourne 2004, ISBN 978-0-521-83378-3 ( online ).

Web links

• LD Kudryavtsev: Monotone function . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Christian Franzki: monotony of functions (the behavior of the function compared to the derivative function). In: Mathematik-Wissen.de. March 6, 2012, accessed May 16, 2015 . 

Individual evidence

1. ↑ Klaus D. Schmidt: Measure and probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 294 , doi : 10.1007 / 978-3-642-21026-6 .