استاد : دکتر گلبابایی in detail this means three conditions: 1. f has to be...

THE CONTINUOUS FUNCTION

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In detail this means three conditions :

1. f has to be defined at c. 2. the limit on the left hand side of that

equation has to exist. 3. the value of this limit must equal f(c).

Definition in terms of limits of sequences

One can instead require that for any sequence of points in the

domain which converges to c.

the corresponding sequence converges to f(c). In mathematical notation

Weierstrass definition (epsilon-delta) of continuous functions

Illustration of the ε-δ-definition: for ε=0.5, c=2 the value δ=0.5 satisfies the condition of the definition.

Given a function f as above and an element c of the domain I, f is said to be continuous at the point c if the following holds: For any number ε > 0, however small, there exists some number δ > 0 such that for all x in the domain of f with c − δ < x < c + δ, the value of f(x) satisfies

Alternatively written, continuity of f : I → R at c ∈ I means that for every ε > 0 there exists a δ > 0 such that for all x ∈ I,:

The following functions are continuous everywhere

PolynomialsRational functionsTrigonometric functionsInverse trigonometric functionsExponential and log functionsPower functions

Directional and semi-continuity

A right-continuous function

A right-continuous function

A left-continuous function

f is said to be right-continuous at the point c if the following holds: For any number ε > 0 however small, there exists some number δ > 0 such that for all x in the domain with c < x < c + δ, the value of f(x) will satisfy

This is the same condition as for continuous functions, except that it is required to hold for x strictly larger than c only. Requiring it instead for all x with c − δ < x < c yields the notion of left-continuous functions. A function is continuous if and only if it is both right-continuous and left-continuous.

Definition : Let f be a function with domain D < R and let a be an element of D. We will say that f is continuous at a if, for each ϵ > 0, there isa > 0 , such that

whenever xЄ D and

Example: Prove that f(x) = x^2 is continuous at x = 2.

We have Solution:

If we insist that , then 1 < x < 3 and so

. Thus, given ϵ > 0, if we choose, then

Whenever

This proves that f is continuous at 2.

uniformly continuous

Example uniformly continuous

Uniform continuity

Definition for functions on metric spacesGiven metric spaces (X, d1) and (Y, d2), a function f : X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d1(x, y) < δ, we have that d2(f(x), f(y)) < ε.If X and Y are subsets of the real numbers, d1 and d2 can be the standard Euclidean norm, || · ||, yielding the definition: for all ε > 0 there exists a δ > 0 such that for all x, y ∈ X, |x − y| < δ implies |f(x) − f(y)| < ε.The difference between being uniformly continuous, and simply being continuous at every point, is that in uniform continuity the value of δ depends only on ε and not on the point in the domain

Discontinuity of the first and second

HOLDER CONDITIO

N

One-sided Lipschitz

Let F(x) be an upper semi-continuous function of x, and that F(x) is a closed, convex set for all x. Then F is one-sided Lipschitz .if

for some C for all x1 and x2

Pointwise Holder continuity

Lipschitz continuity

The function f(x) = √x² + 5 defined for all real numbers is Lipschitz continuous with the Lipschitz constant K =1

Uniform and Local holder continuity :

Holder Continuous / Holder Continuity

Example: F(x)=√x defined on [0,1] is Holder continuous for α ≤

1/2 .

Lipschitz Condition and continuous

Lipschitz Functions

Definition: Let f (x) be defined on an interval I and suppose we can find two positive constants M and a such t hat

Then f is said to satisfy a Lipschitz Condition o f order a and we say that f Lip (a )∈.

Example: Take f (x) = x on the interval [a, b ]. Then

That implies that f Lip (1). Now take f (x) = x∈ on the interval [a, b ]. Then

with M = 2 max(|a|, |b |). Hence, again f Lip (1). The function f ∈(x) = 1 /x on (0, 1). Is it Lip (1) ? How about Lip (1/2)? How about Lip (a )?

Lipschitz and Continuity

Theorem: If f Lip (a ) on I, then f is continuous; indeed, ∈uniformly continuous o n I.

Last time we did continuity with ε and δ An altrnative definition of continuity familar from calculus is: f is continuous at x = c if:

If 0 < α ≤ β ≤ 1 then all Hölder continuous functions on a bounded set Ω are also Hölder continuous. This also includes β = 1 and therefore all lipschitz continuous functions on a bounded set are also C0, α Hölder continuous. The function defined on [0, 1] is not Lipschitz continuous, but is C0, α Hölder continuous for α ½ ≤In the same manner, the function f (x) = xβ (with β ≤ 1) defined on [0, 1] serves as a prototypical example of a function that is C0, α Hölder continuous for 0 < α ≤ β, but not for α > β. There are examples of uniformly continuous functions that are not α–Hölder continuous for any α. For instance, the function defined on [0, 1/2] by f(0) = 0 and by f(x) = 1 / log(x) otherwise is continuous, and therefore uniformly continuous by theHeine-Cantor theorem. It does not satisfy a Hölder condition of any order, however. For α > 1, any α–Hölder continuous function on [0, 1] is a constant.

absolutely continuous

DefinitionLet I be an interval in the real line R. A function f: I → R is absolutely continuous on I if for every positive number ε,

there is a positive number δ.

such that whenever a finite sequence of pair wise disjoint sub-intervals (xk, yk) of I satisfies

then

Equivalent definitions

The following conditions on a real-valued function f on a compact interval [a,b] are equivalent:(1) f is absolutely continuous;(2) f has a derivative f ′ almost everywhere, the

derivative is Lebesgue integrable, and

for all x on[a,b] ;) 3 (there exists a Lebesgue integrable function g on [a,b] such that

for all x on [a,b].If these equivalent conditions are satisfied then necessarily g = f almost everywhere.′

for all x on [a,b]

Every lipschitz - continuous function is absolutely continuous

Every absolutely continuous function is uniformly continuous and, therefore, continuous

Examples

The following functions are continuous everywhere but not absolutely continuous:•the cantor function; •the function

on a finite interval containing the origin;

Generalizations

Let (X, d) be a metric space and let I be an interval in the real line R. A function f: I → X is absolutely continuous on I if for every positive number ε, there is a positive number δ

such that whenever a finite sequence of pairwise disjointsub-intervals [xk, yk] of I satisfies

Absolute continuity of measures

A measure μ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure λ (in other words, dominated by λ) if μ(A) = 0 for every set A for which λ(A) = 0. This is written as “μ << λ”.In most applications, if a measure on the real line is simply said to be absolutely continuous — without specifying with respect to which other measure it is absolutely continuous — then absolute continuity with respect to Lebesgue measure is meant.The same holds for Rn for all n=1,2,3,

CAUCHY-CONTINUOUS FUNCTION

Let X and Y be metric space, and let f be a function from X to Y. Then f is Cauchy-continuous if and only if, given any Cauchy sequence (x1, x2, …) in X, the sequence (f(x1),

f(x2), …) is a Cauchy sequence in Y.

equicontinuous

Let X and Y be two metric space, and F a family of functions from X to Y.The family F is equicontinuous at a point x0 ∈ X if for every ε > 0, there exists a δ > 0 such that d (ƒ(x0), ƒ(x)) < ε for all ƒ ∈ F and all x such that d(x0, x) < δ. The family is equicontinuous if it is equicontinuous at each point of X.The family F is uniformly equicontinuous if for every ε > 0, there exists a δ > 0 such that d(ƒ(x1), ƒ(x2)) < ε for all ƒ ∈ F and all x1, x2 ∈ X such that d(x1, x2) < δ.[

For comparison, the statement 'all functions ƒ in F are continuous' means that for every ε > 0, every ƒ ∈ F, and every x0 ∈ X, there exists a δ > 0 such that d(ƒ(x0), ƒ(x)) < ε for all x ∈ X such that d(x0, x) < δ.For continuity, δ may depend on ε, x0 and ƒ.For uniform continuity, δ may depend on ε, and ƒ.For equicontinuity, δ may depend on ε, and x0.For uniform equicontinuity, δ may solely depend on ε.

Every Lipschitz continuous map between two metric spaces is uniformly continuous. In particular, every function which is differentiable and has bounded derivative is uniformly continuous. More generally, every Holder continuous function is uniformly continuous.Every member of a uniformly equi continuous set of functions is uniformly continuous.The tangent function is continuous on the interval (−π/2, π/2) but is not uniformly continuous on that interval.    ex is continuous everywhere

on the real line but is not uniformly continuous on the line.The exponential function x→ 

Uniformly Continuous continuous

If f is continuous on [a, b], then f is uniformly continuous on [a, b].

If f is a Continuous functions of a compact set ,it is uniformly Continuous

on that set.

uniformly continuous and not lipschitz continuous

Every lipschitz continuous map is uniformlycontinuous

F(x)=√x on [0,1]Converse Example

uniformly continuous functions that are not α–Holder continuous

Any Holder continuous function is uniformly continuous

There are examples of uniformly continuous functions that are not α–Holder continuous for any α. For instance, the function defined on [0, 1/2] by f (0) = 0 and by f(x) = 1 / log(x) otherwise is continuous, and therefore uniformly continuous by the Heine-Cantor theorem. It does not satisfy a Holder condition of any order.

Absolute continuity implies uniform continuity

For a uniformly continuous function f defined on an interval of the real line, if

it is piecewise convex, then it is also absolutely continuous.

Every uniformly continuous function is also Cauchy-continuous

uniformly continuous function Cauchy-continuous

For example, define a two-valued function so that f(x) is 0 when x2 is less than 2 but 1 when x2 is greater than 2. (Note that x2 is never equal to 2 for any rational number x.) This function is continuous on but not ℚCauchy-continuous, since it can't be extended to . On the other hand, ℝany uniformly continuous function on must be Cauchy-continuous. ℚFor a non-uniform example on , let ℚ f(x) be 2x; this is not uniformly continuous (on all of ), but it is Cauchy-continuousℚ.

Let f satisfies |f(x+u)−f(x)|≤L|u|^ α for some constants L and α . If α=1 then f is called Lipschitz continuous, and if0<α<1 then f is Hölder continuous

Lipschitz and holder continuous

A generalization of Lipschitz continuity is called holder continuity

If f is Lipschitz continuity ,then f is absolutely

Every member of a uniformly equicontinuous set of function s is uniformly continuous

A set of functions with the same lipschitz constant is .(uniformly) equicontinuous

For linear transformations f:V→W,uniform continuity is equivalent to continuity.

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