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1.7: Limits, Continuity, and Differentiability
In mathematics , a continuous function is a function that does not have any abrupt changes in value , known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon—delta definition were made to formalize it.
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. A stronger form of continuity is uniform continuity.
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, the function H t denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M t denoting the amount of money in a bank account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.
A form of the epsilon—delta definition of continuity was first given by Bernard Bolzano in Cours d'Analyse , p. Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today see microcontinuity.
The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the s but the work wasn't published until the s. All three of those nonequivalent definitions of pointwise continuity are still in use.
A real function , that is a function from real numbers to real numbers, can be represented by a graph in the Cartesian plane ; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. A more mathematically rigorous definition is given below. A rigorous definition of continuity of real functions is usually given in a first course in calculus in terms of the idea of a limit. First, a function f with variable x is said to be continuous at the point c on the real line, if the limit of f x , as x approaches that point c , is equal to the value f c ; and second, the function as a whole is said to be continuous , if it is continuous at every point.
A function is said to be discontinuous or to have a discontinuity at some point when it is not continuous there. These points themselves are also addressed as discontinuities. There are several different definitions of continuity of a function. Sometimes a function is said to be continuous if it is continuous at every point in its domain.
Sometimes an exception is made for boundaries of the domain. In this case only the limit from the right is required to equal the value of the function.
The most common and restrictive definition is that a function is continuous if it is continuous at all real numbers.
In this case, the previous two examples are not continuous, but every polynomial function is continuous, as are the sine , cosine , and exponential functions. Care should be exercised in using the word continuous , so that it is clear from the context which meaning of the word is intended. Using mathematical notation, there are several ways to define continuous functions in each of the three senses mentioned above.
Some possible choices include. The function f is continuous at some point c of its domain if the limit of f x , as x approaches c through the domain of f , exists and is equal to f c. In detail this means three conditions: first, f has to be defined at c guaranteed by the requirement that c is in the domain of f.
Second, the limit on the left hand side of that equation has to exist. Third, the value of this limit must equal f c. The formal definition of a limit implies that every function is continuous at every isolated point of its domain.
A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c. Intuitively, a function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point f c as the width of the neighborhood around c shrinks to zero.
This definition only requires that the domain and the codomain are topological spaces and is thus the most general definition.
It follows from this definition that a function f is automatically continuous at every isolated point of its domain.
As a specific example, every real valued function on the set of integers is continuous. More intuitively, we can say that if we want to get all the f x values to stay in some small neighborhood around f x 0 , we simply need to choose a small enough neighborhood for the x values around x 0. In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology , here the metric topology.
In proofs and numerical analysis we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalise this to a definition of continuity.
A function is continuous in x 0 if it is C -continuous for some control function C. This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.
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 Non-standard analysis is a way of making this mathematically rigorous.
The real line is augmented by the addition of infinite and infinitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be defined as follows. In other words, an infinitesimal increment of the independent variable always produces to an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy 's definition of continuity.
Checking the continuity of a given function can be simplified by checking one of the above defining properties for the building blocks of the given function. It is straightforward to show that the sum of two functions, continuous on some domain, is also continuous on this domain. Thus it is a continuous function. However, unlike the previous example, G can be extended to a continuous function on all real numbers, by defining the value G 0 to be 1, which is the limit of G x , when x approaches 0, i.
The term removable singularity is used in such cases, when re defining values of a function to coincide with the appropriate limits make a function continuous at specific points. A more involved construction of continuous functions is the function composition. Given two continuous functions. Intuitively we can think of this type of discontinuity as a sudden jump in function values. Yet another example: the function. Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined pathological , for example, Thomae's function ,.
In a similar vein, Dirichlet's function , the indicator function for the set of rational numbers,. The intermediate value theorem is an existence theorem , based on the real number property of completeness , and states:. The extreme value theorem states that if a function f is defined on a closed interval [ a , b ] or any closed and bounded set and is continuous there, then the function attains its maximum, i.
The same is true of the minimum of f. Every differentiable function. The converse does not hold: for example, the absolute value function. Weierstrass's function is also everywhere continuous but nowhere differentiable. More generally, the set of functions. See differentiability class. In the field of computer graphics, properties related but not identical to C 0 , C 1 , C 2 are sometimes called G 0 continuity of position , G 1 continuity of tangency , and G 2 continuity of curvature ; see Smoothness of curves and surfaces.
The converse does not hold, as the integrable, but discontinuous sign function shows. Given a sequence. The pointwise limit function need not be continuous, even if all functions f n are continuous, as the animation at the right shows.
However, f is continuous if all functions f n are continuous and the sequence converges uniformly , by the uniform convergence theorem. This theorem can be used to show that the exponential functions , logarithms , square root function, and trigonometric functions are continuous.
Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity or right and left continuous functions and semi-continuity.
Roughly speaking, a function is right-continuous if no jump occurs when the limit point is approached from the right. This is the same condition as for continuous functions, except that it is required to hold for x strictly larger than c only.
A function is continuous if and only if it is both right-continuous and left-continuous. A function f is lower semi-continuous if, roughly, any jumps that might occur only go down, but not up. The reverse condition is upper semi-continuity. The concept of continuous real-valued functions can be generalized to functions between metric spaces. A metric space is a set X equipped with a function called metric d X , that can be thought of as a measurement of the distance of any two elements in X.
Formally, the metric is a function. Given two metric spaces X , d X and Y , d Y and a function. The latter condition can be weakened as follows: f is continuous at the point c if and only if for every convergent sequence x n in X with limit c , the sequence f x n is a Cauchy sequence , and c is in the domain of f.
This notion of continuity is applied, for example, in functional analysis. A key statement in this area says that a linear operator. Thus, any uniformly continuous function is continuous.
The converse does not hold in general, but holds when the domain space X is compact. Uniformly continuous maps can be defined in the more general situation of uniform spaces. That is, a function is Lipschitz continuous if there is a constant K such that the inequality. Another, more abstract, notion of continuity is continuity of functions between topological spaces in which there generally is no formal notion of distance, as there is in the case of metric spaces. A topological space is a set X together with a topology on X , which is a set of subsets of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point.
The elements of a topology are called open subsets of X with respect to the topology.
The concept of the limit is one of the most crucial things to understand in order to prepare for calculus. A limit is a number that a function approaches as the independent variable of the function approaches a given value. In the following sections, we will more carefully define a limit, as well as give examples of limits of functions to help clarify the concept. Continuity is another far-reaching concept in calculus. A function can either be continuous or discontinuous. One easy way to test for the continuity of a function is to see whether the graph of a function can be traced with a pen without lifting the pen from the paper.
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To develop calculus for functions of one variable, we needed to make sense of the concept of a limit, which we needed to understand continuous functions and to define the derivative. Limits involving functions of two variables can be considerably more difficult to deal with; fortunately, most of the functions we encounter are fairly easy to understand. Sadly, no.
In mathematics , a continuous function is a function that does not have any abrupt changes in value , known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous.