Steadiness

the steadiness is a concept of mathematics, which is particularly in the subsections of the analysis and the topology of central importance. A function constantly means, if infinitesimal small changes of the argument (the arguments) only to infinitesimal smallChanges of the function value lead. That means in particular that in the function values no jumps arise. The opposite of constant is unstable.

Table of contents

definitions

Graphische Veranschaulichung einer unstetigen reellen Funktion
graphic illustration of an unstable real function

the idea of the steadiness can be described as follows:

A reellwertige function< math> f: I \ tons \ mathbb {R}< /math>on a real interval <math> I \ subseteq \ mathbb {R}< /math> is constant, if the graph of the function <math> f< /math> without setting the pin off to be drawn can. The function may have in particular no saltuses.

This statement is not a definition, because is unclear, which under in a coursedraw exactly to understand is, for example with a curve, which has an infinite length on a finite interval. Nevertheless it corresponds approximately to the meaning of the steadiness and is very useful therefore for the opinion. The following definitions for the steadinessare mathematically accurate.

Augustin Louis Cauchy and Berne pool of broadcasting corporations Bolzano gave to at the beginning 19. Century independently a definition of steadiness. They constantly called a function, if arbitrarily small changes of the argument drew only arbitrarily small changes of the function value.This was already an accurate definition, which leaves certain questions open however in their practical application. The definition used nowadays comes from Karl Weierstrass from the end 19. Century. This math \ <epsilon> \ delta in such a way specified< /math> - criterion implements the arbitrarily small changes more exactly.

it is characterised steadiness of real

functions real functions by the fact that its definition range <math> D< /math> and their target area subsets of the real numbers are. For such functions <math> f </math> the steadiness is in one point <math> x_0 </math> of theDefinition range defines as follows:

<math> f \ colon D \ tons \ R< /math> is constant in <math> x_0 \ in D </math> exactly if
for all <math> \ varepsilon > 0< /math> <math> \ a delta > 0< /math> existed, so that for all <math> x \ in D< /math> with <math>|x - x_0| < \ delta< /math> applies:<math>|f (x) - f (x_0)| < \ varepsilon </math>.

Equivalent one in addition is the following definition:

<math> f \ colon D \ tons \ R \ mbox {constantly in} x_0 \ Longleftrightarrow \ lim_ {x \ rightarrow x_0} f (x) =f (x_0) </math>

A function constantly means, if it is constant in each place of their definition range.

Z. B.the victory around function is

<math> \ operator name {sgn} (x) = \ begin {cases} 1 & x> 0 \ \ 0 & x=0 \ \ -1 & x< 0 \ ends {cases}< /math>

in each place <math> x \ in \ R \ set minus \ {0 \}< /math> constantly, but not altogether constantly, there it in the place <math> 0< /math> is unstable: The left-sided limit value is -1, the rechtsseitigen limit value+1 is and thus exists the limit value <math> \ lim_ {to x \ tons of 0} \, \ operator name {sgn} (x)< /math> not.

characteristics

  • are <math> f </math> and <math> g </math> constantly with a common definition range, then are also <math> f + g </math>, <math> f - g</math> and <math> f \ cdot g </math> constantly. Math <g> (x) is \ ne 0< /math> for all <math> x </math> in the definition range, then is also <math> \ frac {f} {g} </math> constantly.
  • The composition <math> f \ circ g </math> two constant functions is likewise constant.
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Examples

  • the sine function <math> \ sin: \ R \ tons \ R, \; x \ mapsto \ sin (x) </math> is constant (D. h. in particular in each point <math> x \ in \ R< /math> constantly).
  • The cosine function <math> \ cos: \ R \ tons \ R, \; x \ mapsto \ cos (x) </math> is constant.
  • <math> f:\R\to\R, \; x \ mapsto e^ {\ cos (x)} </math> is (as composition of the exponential and thatCosine function) constantly.
  • The function <math> f: D \ tons \ mathbb {R}, \ x \ mapsto \ frac {1} {x}< /math> D= is \ mathbb { <R>} \ set minus \ {0 \} /math on the maximum definition range< math> constantly. In the place 0 the term steadiness is not applicable and <math> f </math> is neither constant nor unstable in 0.
  • The tangent function< math> \ tan (x)= \ frac {\ sin (x)}{\ cos (x)} </math> D is constant in its definition range. h. in all <math> x </math> out <math> \ R< /math> with <math> \ cos (x) \ neq 0< /math>.
  • The complex exponential function <math> \ Bbb C \ tons \ Bbb C, \; z \ mapsto \ exp (z) </math> is constant.

Verallgemeinerung: Constant functions between metric areas

a function constantly means, if their function value changes sufficient little, as long as one sufficient little changes only the function argument. Also this is only one description, possible accurate definitions is the following:

Epsilon delta criterion

Are <math> (X, d_X) </math>, <math> (Y, d_Y) </math>metric areas. A function <math> f: X \ rightarrow Y </math> is constantly called in <math> x_0< /math>, if applies

< math>
\ forall \ epsilon  > 0\; \ exists \ delta  > 0 \; \ forall x \ in U_ \ delta (x_0): d_Y (f (x), f (x_0))  < \ epsilon

< /math>

Designates <math> U_ {\ delta} (x_0) = \ {x \ in X | d_X (x, x_0) < \ delta \} </math> the open <math> \ delta< /math> - environment over <math> x_0 </math>.

consequence criterion

are <math> (X, d_X) </math>, <math> (Y, d_Y) </math> metric areas, then applies:

<math> f: X \ tons of Y </math> is constant in <math> x_0 \ Leftrightarrow< /math> For each consequence <math> (x_n) </math> from the definition quantity of <math> f </math>, approximately <math> x_0 </math> converged, converges <math> f (x_n) </math> approximately <math> f (x_0) </math>.

environment criterion

are <math> (X, d_X) </math>, <math> (Y, d_Y) </math> metric areas, then applies:

<math> f: X \ tons of Y </math> is constant in <math> x_0 \ Leftrightarrow< /math> To each environment <math> V </math> of <math> f (x_0) </math> there is oneEnvironment <math> U </math> of <math> x_0 </math> , so that for all <math> x \ isin U \ cap X </math> applies: <math> f (x) \ isin V </math>.

further Verallgemeinerung: Constant functions between topological areas

major items: Steadiness (topology)

all pastDefinitions are specializations of the appropriate definition of steadiness in the topology. There a function between two topological areas is exactly constant if the Urbilder of open quantities is again open quantities.

special cases of steadiness

of special cases thatSteadiness are z. B.even steadiness, (local) Lipschitz steadiness as well as absolute steadiness. The usual steadiness is called every now and then also point for point steadiness, in order to define it in relation to the even steadiness. Applications of the Lipschitz steadiness are to z. B. in uniqueness conditions (z.B.Sentence of Picard Lindelöf) for initial value problems. The absolute steadiness finds use in the stochastics and the masstheorie.

A characteristic, which can possess a quantity of functions, is the gleichgradige steadiness. It plays a role in frequentused sentence of Arzelà Ascoli.

the following

connections in the case of real functions apply for connection:

<math> f< /math> Lipschitz constantly <math> \ Rightarrow< /math> <math> f< /math> locally Lipschitz constantly <math> \ Rightarrow< /math> <math> f< /math> constantly

and

< math> f< /math> Lipschitz constantly <math> \ Rightarrow< /math> <math> f< /math> absolutely constantly <math> \ Rightarrow< /math> <math> f< /math> evenly constantly <math> \ Rightarrow< /math> <math> f< /math> constantly.

examples

some counter examples are to demonstrate that the back directions in all rule do not apply:

  • <math> f: [- 1.1] \ rightarrow \ R, x \ mapsto \ sqrt [3] {x}< /math> is constant, but not locally Lipschitz constant.

important sentences over constant functions

concatenation of constant functions

each concatenation of constant functions are also again constant.

steadiness of the inverse function

are <math> I< /math> an interval in <math> \ mathbb {R}< /math> and <math> f \ colon I \ rightarrow \ mathbb R< /math> a constant, strictlymonotonous growing function, then is the picture of <math> f< /math> an interval <math> J< /math>, <math> f \ colon I \ tons of J< /math> , and the inverse function is bijektiv <math> f^ {- 1} \ colon J \ tons of I< /math> is constant. Thus is <math> f< /math> a Homöomorphismus of <math> I< /math> after <math> J< /math>.

This applies as indicated onlyfor functions, which are constant in the entire interval. Is <math> f< /math> a reversible and in the place <math> x_0< /math> constant function, then is the inverse function <math> f^ {- 1}< /math> in the place <math> f^ {- 1} (x_0) </math> generally not constantly. As counter example is <math> f< /math> defined through:

  • on <math> (2C, 2c+1)< /math> math <f> (x) is =x-k< /math> (<math> k< /math> goes through the positive whole numbers)
  • on <math> (2k-1,2k)< /math> math <f> (x) is = \ frac {1} {x}< /math>
  • on <math> \ (\ frac1 {k+1}, \ frac1k \ right) /math< left> math <f> (x) is = \ frac1 {\ frac1x+k}< /math>
  • <math> f (0) =0< /math>, <math> f (k) =k </math>, <math> f \ left (\ frac {1} {k} \ right) = \ frac {1} {k}< /math>
  • <math> f (x) =-f (- x)< /math> for <math> x< 0< /math>.

Then is <math> f< /math> bijektiv and in 0 constantly, but <math> f^ {- 1}< /math> is unstable in 0.

the intermediate value set

the intermediate value set mentioned that one on the interval <math> [A, b]< /math> (with <math> A< b< /math>) constant function each function value between <math> f (A)< /math> and <math> f (b)< /math> at least once assumes.

Formally:

Is <math> f: [A, b] \ tons \ mathbb {R}< /math> a constant function with <math> A< b< /math> and <math> f (A)< f (b)< /math>, then exists for all<math> D \ in [f (A), f (b)]</math> <math> an x \ in [A, b]< /math>, so that <math> f (x) =d< /math>.
Similar for <math> f (A)> f (b)< /math> and <math> D \ in [f (b), f (A)]</math>.

An equivalent formulation is: The picture of a constant function on an interval is interval. (The picture of an open or half-open interval knows however quite an final intervalits.)

consequence convergence of constant real who tigers functions

is <math> f </math> a reellwertige function, those on their definition range< math> D (f) </math> is constant, <math> D (f) </math> is a subset of the real numbers, <math> x_0 </math> is from the definition rangeof <math> f </math>,

then applies to each consequence of real numbers <math> x_n </math> out <math> D (f) </math> approximately <math> x_0 </math> converged that the consequence of the function values <math> f (x_n) </math> approximately <math> f (x_0) </math> converged.

Note:ThisSentence applies also to constant illustrations between arbitrary metric areas.

sentence of Bolzano

takes the function constant on an final interval <math> to f (x)< /math> in two places <math> A </math> and <math> b </math> this interval of function values with differentSign on, then gives it between <math> A </math> and <math> b </math> at least one place <math> C </math>, at that the function <math> f (x)< /math> disappears (D. h. <math> f (C) =0< /math> thus a zero of the function).

sentence of Weierstrass

onereellwertige function, those on an final and limited subset of <math> \ mathbb R^n< /math> , assumes its upper and its lower limit is constant. For real functions that can be reformulated as follows: Math <f> is \ colon [A, b] \ ton \ mathbb {R}< /math> constantly, then there is places <math> t, h \ in [A, b]< /math>, thusthat

<math> f (t) \ leq f (x) \ leq f (h)< /math> for all <math> x \ in [A, b]< /math>

applies.

The sentence of Weierstrass needs fewer conditions for the search for high and Tiefpunkten (see extreme value) of a function than the differenzielle search.

differentiability of constant functions

constant functions are not necessarily differentiably. Still at the beginning 19. Century was convinced one that a constant function could not be differentiable at the most in few places (like the amount function). Bernhard Bolzano designed then as first mathematicians actually a function, those everywhere constantly,but is not differentiable anywhere, which does not admit in the professional world however became;Karl Weierstrass found then into the 1860ern likewise a such function, which struck waves this time among mathematicians. Its function is as follows defined

< math> f (x) = \ sum_ {j=0} ^ {\ infty} b^n \ cos (a^n \ pi x)< /math>,

how A an odd number is and <math> b \ in (0,1)< /math> with <math> starting from> 2+3 \ pi /2< /math>. A well-known example of a constant, not differentiable function is the cook curve presented of Helge of cook 1904.

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