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From Wikipedia

Classification of discontinuities

Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function.

This article describes the classification of discontinuities in the simplest case of functions of a single real variable taking real values.

Classification of discontinuities

Consider a real valued function ƒ of a real variable x, defined in a neighborhood of the point x0 in which ƒ is discontinuous. Then three situations may be distinguished:

  1. The one-sided limit from the negative direction
    L^{-}=\lim_{x\rarr x_0^{-}} f(x)

    and the one-sided limit from the positive direction

    L^{+}=\lim_{x\rarr x_0^{+}} f(x)

    at x_0 exist, are finite, and are equal to L=L^{-}=L^{+}. Then, if ƒ(x0) is not equal to L, x0 is called a removable discontinuity. This discontinuity can be 'removed to make ƒ continuous at x0', or more precisely, the function

    g(x) = \begin{cases}f(x) & x\ne x_0 \\ L & x = x_0\end{cases}

    is continuous at x=x0.

  2. The limits L^{-} and L^{+} exist and are finite, but not equal. Then, x0 is called a jump discontinuity or step discontinuity. For this type of discontinuity, the function ƒ may have any value in x0.
  3. One or both of the limits L^{-} and L^{+} does not exist or is infinite. Then, x0 is called an essential discontinuity, or infinite discontinuity. (This is distinct from the term essential singularitywhich is used when studyingfunctions of complex variables.)

The term removable discontinuity is sometimes incorrectly used for cases in which the limits in both directions exist and are equal, while the function is undefined at the point x_0. This use is improper because continuity and discontinuity of a function are concepts defined only for points in the function's domain. Such a point not in the domain, is properly named a removable singularity.

The oscillation of a function at a point quantifies these discontinuities as follows:

  • in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
  • in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits from the two sides);
  • in an essential discontinuity, oscillation measures the failure of a limit to exist.

Examples

1. Consider the function

f(x)=\begin{cases}x^2 & \mbox{ for } x< 1 \\ 0 & \mbox { for } x=1 \\ 2-x& \mbox{ for } x>1\end{cases}

Then, the point x_0=1 is a removable discontinuity.

2. Consider the function

f(x)=\begin{cases}x^2 & \mbox{ for } x< 1 \\ 0 & \mbox { for } x=1 \\ 2-(x-1)^2& \mbox{ for } x>1\end{cases}

Then, the point x_0=1 is a jump discontinuity.

3. Consider the function

f(x)=\begin{cases}\sin\frac{5}{x-1} & \mbox{ for } x< 1 \\ 0 & \mbox { for } x=1 \\ \frac{0.1}{x-1}& \mbox{ for } x>1\end{cases}

Then, the point x_0=1 is an essential discontinuity (sometimes called infinite discontinuity). For it to be an essential discontinuity, it would have sufficed that only one of the two one-sided limits did not exist or were infinite. However, given this example the discontinuity is also an essential discontinuity for the extension of the function into complex variables.

The set of discontinuities of a function

The set of points at which a function is continuous is always a Gδ set. The set of discontinuities is an Fσ set.

The set of discontinuities of a monotonic function is at most countable. This is Froda's theorem.

Thomae's functionis discontinuous at every rational point, but continuous at every irrational point.

The indicator function of the rationals, also known as the Dirichlet function, isdiscontinuous everywhere.


Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".

Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.

As an example, consider the function h(t) which describes the height of a growing flower at time t. This function is continuous. In fact, there is a dictum of classical physics which states that in nature everything is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous. (However, if one assumes a discrete set as the domain of function M, for instance the set of points of time at 4:00 PM on business days, then M becomes continuous function, as every function whose domain is a discrete subset of reals is.)

Real-valued continuous functions

Historical infinitesimal definition

Cauchy defined continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34).

Definition in terms of limits

Suppose we have a function that maps real numbers to real numbers and whose domain is some interval, like the functions h and M above. Such a function can be represented by a graph in the Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps".

In general, we say that the function f is continuous at some pointc of its domain if, and only if, the following holds:

  • The limit of f(x) as x approaches c through domain of f does exist and is equal to f(c); in mathematical notation, \lim_{x \to c}{f(x)} = f(c). If the point c in the domain of f is not a limit point of the domain, then this condition is vacuously true, since x cannot approach c through values not equal c. Thus, for example, every function whose domain is the set of all integers is continuous.

We call a function continuous if and only if it is continuous at every point of its domain. More generally, we say that a function is continuous on some subset of its domain if it is continuous at every point of that subset.

The notation C(Ω) or C0(Ω) is sometimes used to denote the set of all continuous functions with domain Ω. Similarly, C1(Ω) is used to denote the set of differentiable functions whose derivative is continuous, C²(Ω) for the twice-differentiable functions whose second derivative is continuous, and so on (see differentiability class). In the field of computer graphics, these three levels are sometimes called g0 (continuity of position), g1 (continuity of tangency), and g2 (continuity of curvature). The notation C(n, α)(Ω) occurs in the definition of a more subtle concept, that of Hölder continuity.

Weierstrass definition (epsilon-delta) of continuous functions

Without resorting to limits, one can define continuity of real functions as follows.

Again consider a function ƒ that maps a set of real numbers to another set of real numbers, and suppose c is an element of the domain of ƒ. The function ƒ is said to be continuous at the point c if the following holds: For any number ε&nbsp;>&nbsp;0, however small, there exists some number δ&nbsp;>&nbsp;0 such that for all x in the domain of ƒ with c&nbsp;&minus;&nbsp;δ&nbsp;<&nbsp;x&nbsp;<&nbsp;c&nbsp;+&nbsp;δ, the value of ƒ(x) satisfies

f(c) - \varepsilon < f(x) < f(c) + \varepsilon.\,

Alternatively written: Given subsets I, D of R, continuity of ƒ&nbsp;:&nbsp;I&nbsp;→&nbsp;D at c&nbsp;∈&nbsp;I means that for every&nbsp;ε&nbsp;>&nbsp;0 there exists a δ&nbsp;>&nbsp;0 such that for all x&nbsp;∈&nbsp;I,:

| x - c | < \delta \Rightarrow | f(x) - f(c) | < \varepsilon. \,

A form of this epsilon-delta definition of continuity was first given by Bernard Bolzano in 1817. Preliminary forms of a related definition of the limit were given by Cauchy, though the formal definition and the distinction between poin

Absolute continuity

In mathematics, the relationship between the two central operations of calculus, differentiation and integration, stated by fundamental theorem of calculus in the framework of Riemann integration, is generalized in several directions, using Lebesgue integration and absolute continuity. For real-valued functions on the real line two interrelated notions appear, absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to Radon–Nikodym derivative, or density, of a measure.

Absolute continuity of functions

It may happen that a continuous function f is differentiable almost everywhere on [0,1], its derivative f&nbsp;′ is Lebesgue integrable, and nevertheless the integral of f&nbsp;′ differs from the increment of f. For example, this happens for the Cantor function, which means that this function is not absolutely continuous. Absolute continuity of functions is a smoothness property which is stricter than continuity and uniform continuity.

Definition

Let I be an interval in the real lineR. A function f: I→ R is absolutely continuous on I if for every positive number \epsilon, there is a positive number \delta such that whenever a finite sequence of pairwise disjoint sub-intervals (xk, yk) of I satisfies

\sum_{k} \left| y_k - x_k \right| < \delta

then

\displaystyle \sum_{k} | f(y_k) - f(x_k) | < \epsilon.

The collection of all absolutely continuous functions on I is denoted AC(I).

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&nbsp;′ almost everywhere, the derivative is Lebesgue integrable, and
f(x) = f(a) + \int_a^x f'(t) \, dt
for all x on [a,b];
(3) there exists a Lebesgue integrable function g on [a,b] such that
f(x) = f(a) + \int_a^x g(t) \, dt
for all x on [a,b].

If these equivalent conditions are satisfied then necessarily g = f&nbsp;′ almost everywhere.

Equivalence between (1) and (3) is known as the fundamental theorem of Lebesgue integral calculus, due to Lebesgue.

For an equivalent definition in terms of measures see the section Relation between the two notions of absolute continuity.

Properties

  • The sum and difference of two absolutely continuous functions are also absolutely continuous. If the two functions are defined on a bounded closed interval, then their product is also absolutely continuous.
  • If an absolutely continuous function is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely continuous.
  • Every absolutely continuous function is uniformly continuous and, therefore, continuous. Every Lipschitz-continuousfunction is absolutely continuous.
  • If f: [a,b] → R is absolutely continuous, then it is of bounded variation on [a,b].
  • If f: [a,b] → R is absolutely continuous, then it has the Luzin N property (that is, for any L \subseteq [a,b] such that \lambda(L)=0, it holds that \lambda(f(L))=0, where \lambda stands for the Lebesgue measure on R).
  • f: I→ R is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin N property.

Examples

The following functions are continuous everywhere but not absolutely continuous:

f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ x \sin(1/x), & \mbox{if } x \neq 0 \end{cases}
on a finite interval containing the origin;
  • the function Æ’(x) = x&nbsp;2 on an unbounded interval.

Generalizations

Let (X, d) be a metric space and let I be an interval in the real lineR. A function f: I→ X is absolutely continuous on I if for every positive number \epsilon, there is a positive number \delta such that whenever a finite sequence of pairwise disjoint sub-intervals [xk, yk] of I satisfies

\sum_{k} \left| y_k - x_k \right| < \delta

then

\sum_{k} d \left( f(y_k), f(x_k) \right) < \epsilon.

The collection of all absolutely continuous functions from I into X is denoted AC(I; X).

A further generalization is the space ACp(I; X) of curves f: I→ X such that

d \left( f(s), f(t) \right) \leq \int_{s}^{t} m(\tau) \, \mathrm{d} \tau \mbox{ for all } [s, t] \subseteq I

for some m


From Yahoo Answers

Question:so what if i have a point discontinuity how will that affect my answer for limit? for example (this isn't point) what is the answer for 1/x+3 as limit x->-3? this is so confusing i know what the discontinuities are and know what a limit is but how do they correlate? if its a jjump i know it is going to be a nonremovable discontinuity but how do i find the limit for it?

Answers:Limit: in order to have limit to exits, left side limit AND right side limit has to be equal each other. A point does not have to be defined at that point to find the limit. For example you have. 1/(x+3). You know that you can't plug in x = -3 because 1/(x+3) will be undefined. You want to examine the limit at x = -3; this means what happens AROUND y when some x value is very close to x = -3. If you pick a x value very close to say, x = -3.0001 which is left of -3, you will see that 1/(x+3) = -10000. If you pick a x value that is even closer than x = -3.0001, you will see that it goes to negative infinity. If you pick a x value say, x = -2.9999 which is right of x = -3, 1/(x+3) = 10000; so limit does not exits for this rational expression. Jump function: If it jumps at x = a, limit does not exist at x = a. If it's removable, limit exist. Is that help?

Question:In terms of their graphs and behaviors. I learned it but it slipped my mind. Thanks!

Answers:Continuity requires that the left-hand limit equals the right-hand limit for all x. In other words, the function is defined everywhere and does not "jump" from one value to another or have vertical asymptotes. A function is differentiable if its derivative is defined everywhere. Qualitatively, the function is "smooth" and doesn't have any sharp edges. Note that the derivative may be discontinuous as long as it is defined everywhere. Differentiable functions are a subset of continuous functions.

Question:In your explanation, give examples of the following: a) a function with a nonremovable discontinuity at x=2. b) a function with a removable discontinuity at x= -2. c) a function that has both of the characteristics described in a. and b.

Answers:it's removably discontinuous if you can fix the discontinuity by redifining a finite number of points in the function to fill the hole (the discontinuity) it would be nonremovable if you couldn't create something to fix the hole. it has to do with the limits of the function such as "f(x)=" compared to "f(3)=" a non removable discontinuity occurs when a function has no general limit as the given value (such as x.) there is no way to repair this with a finite number of points.

Question:What is the difference between a differentiable and continuous function? thank you!

Answers:To answer your question visually, a continuous function has no breaks in the curve, while a differentiable function has no breaks in the slope. To give two examples: y = |x| (the absolute value function) is continuous at x = 0, since there is no break in the curve. However, it is not differentiable at x = 0 because that 90 turn it takes. Expressed mathematically, the slope is -1 when you approach it from the left, but +1 when you approach it from the right. y = x /x is not continuous at x = 0 because 0/0 is undefined. However, it is differentiable at x = 0 because the limit approaches the same number (0) from either direction.

From Youtube

Continuous Mind Stuff :For some reason I find it quite difficult to imagine mind and matter as continuous or conterminous, even though this absolutely seems to be the case. I'm sure I'm not the only one, there seems to be quite a bit of evidence the we are all, as Paul Bloom says, 'natural born dualists'. I'm just wondering how this intuitively obvious but factually unlikely belief plays itself out in terms of our understanding and in in terms of application to other aspects of thought and belief. One of the problems, I suspect, is that the dualism is conceived of as some kind of immiscibility of substances (folowing Descartes) and that 'mind stuff' is inherently different from 'body stuff'. This misapprehension may lead to the logical next step, which is the sense that whilst an individual's mind and body are substantially discontinuous from one another there is a potential continuity between one mind and another, or between an individual mind and some kind of panpsychic 'world mind'. I'm not saying this correct of course, just that it feels more intuitive to imagine mind stuff joining up with more of the same mind-stuff that it being supervenient on the matter of the brain/body/world.

Topology #13 Continuity Examples :Examples of continuous and discontinuous functions between topological spaces