Continuity
Continuity in mathematics refers to the property of a function where small changes in the input result in small changes in the output. A function is considered continuous if it can be drawn without lifting the pen from the paper. This concept is fundamental in calculus and analysis, and it helps to understand the behavior of functions and their graphs.
5 Key excerpts on "Continuity"
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 2 Limits and Continuity CHAPTER 2 LIMITS AND Continuity 2.1 INTRODUCTION Limits are the building blocks of calculus. They are used to establish all major concepts, including Continuity, derivatives, and integrals. There are many ways to examine and evaluate limits. Some of the methods that we will review are studying numerical patterns, direct substitution, deducing information from a graph, simplifying prior to substitution, and taking an intuitive approach to the behavior of a particular function. Limits are also closely linked with Continuity. Prior to establishing a formal definition of Continuity, you should recall from previous courses that any kind of a break in the domain of a function is called a disContinuity. Whether a function is continuous or discontinuous at a point in its domain determines the ease with which a limit at that point may be determined. Figure 2.1 Imagine a square with an area of 4 square feet (Figure 2.1). If the consecutive midpoints of each side of the square are connected with segments, a new square is formed with an area of 2 square feet. If the consecutive midpoints of the new square are connected, the resulting area is 1 square foot. Repeating this pattern over and over, the sequence of areas is 4, Even though each new area could continue to be multiplied by forever, it can be said that the limit of the areas is 0. Interestingly, you may recognize this sequence from a previous course as an infinite geometric sequence with a common ratio of The sum of the infinite number of consecutive areas has a limit given by the formula where a is the first term, and r is the common ratio. In this case, square feet. If n is the number of squares whose areas are being summed, the previous conclusion could also be written as 2.2 LIMITS AS X APPROACHES A CONSTANT A NUMERICAL APPROACH Table 2.1 Table 2.1 lists ordered pairs for a certain function, f (x)...
eBook - ePub- Frank J. Fabozzi, Frank J. Fabozzi(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
...The general idea behind Continuity is that the graph of f (x) does not exhibit gaps. In other words, f (x) can be thought of as being seamless. We illustrate this in Figure 1. For increasing x, from x = 0 to x = 2, we can move along the graph of f (x) without ever having to jump. In the figure, the graph is generated by the two functions f (x) = x 2 for x ∈ [0,1), and f (x) = ln(x) + 1 for x ∈ [1, 2). Note that the function f (x) = ln (x) is the natural logarithm. It is the inverse function to the exponential function g (x) = e x where e = 2.7183 is the Euler constant. The inverse has the effect that f (g (x)) = ln(e x) = x, that is, ln and e cancel each other out. A function f (x) is discontinuous if we have to jump when we move along the graph of the function. For example, consider the graph in Figure 2. Approaching x = 1 from the left, we have to jump from f (x) = 1 to f (1) = 0. Thus, the function f is discontinuous at x = 1. Here, f is given by f (x) = x 2 for x ∈ [0,1), and f (x) = ln(x) for x ∈ [1,2). Figure 3 Continuity Criterion Note : Function f = sin(x), for −1 ≤ x ≤ 1. Formal Derivation For a formal treatment of Continuity, we first concentrate on the behavior of f at a particular value x*. We say that that a function f (x) is continuous at x * if, for any positive distance δ, we obtain a related distance ε (δ) such that What does that mean? We use Figure 3 to illustrate. (The function is f (x) = sin(x) with x * = 0.2.) At x *, we have the value f (x *). Now, we select a neighborhood around f (x *) of some arbitrary distance δ as indicated by the dashed horizontal lines through f (x *) − δ and f (x *) + δ, respectively. From the intersections of these horizontal lines and the function graph (solid line), we extend two vertical dash-dotted lines down to the x -axis so that we obtain the two values x L and x U, respectively. Now, we measure the distance between x L and x * and also the distance between x U and x *...
- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...So this is a “bottom-heavy” expression and DID YOU KNOW The sine and cosine functions are fundamental to the theory of periodic functions as those that describe sound and light waves. Continuity Overview: A very loose definition of a continuous curve is one that can be drawn without picking up the pencil from the paper. Lines are continuous, parabolas are continuous, and so are sine curves. But for more complex curves, we need a definition that can prove where a function is continuous. A function f (x) is continuous at x = c if all three conditions hold: What this says is that the limit must exist at x = c, the function must have a value at x = c, and that this limit and value must be the same. If a function is continuous at all values c in its domain, the function is continuous. EXAMPLE 24: For each of the following functions, examine their graphs and determine if the function is continuous at the given value of x and, if not, which of the rules of Continuity above it fails. SOLUTIONS: a. is not continuous at x = 2 because does not exist. b. is not continuous at x = 3 because f (3) does not exist. c. is not continuous at x = 0 because d. is continuous at x = –3. e. is not continuous at x = 1 because does not exist and f (1) does not exist. f. While this...
eBook - ePub- Bertrand Russell(Author)
- 2014(Publication Date)
- Routledge(Publisher)
...The application to actual space and time will not be in question to begin with. I do not see any reason to suppose that the points and instants which mathematicians introduce in dealing with space and time are actual physically existing entities, but I do see reason to suppose that the Continuity of actual space and time may be more or less analogous to mathematical Continuity. The theory of mathematical Continuity is an abstract logical theory, not dependent for its validity upon any properties of actual space and time. What is claimed for it is that, when it is understood, certain characteristics of space and time, previously very hard to analyse, are found not to present any logical difficulty. What we know empirically about space and time is insufficient to enable us to decide between various mathematically possible alternatives, but these alternatives are all fully intelligible and fully adequate to the observed facts. For the present, however, it will be well to forget space and time and the Continuity of sensible change, in order to return to these topics equipped with the weapons provided by the abstract theory of Continuity. Continuity, in mathematics, is a property only possible to a series of terms, i.e. to terms arranged in an order, so that we can say of any two that one comes before the other. Numbers in order of magnitude, the points on a line from left to right, the moments of time from earlier to later, are instances of series. The notion of order, which is here introduced, is one which is not required in the theory of cardinal number. It is possible to know that two classes have the same number of terms without knowing any order in which they are to be taken. We have an instance of this in such a case as English husbands and English wives: we can see that there must be the same number of husbands as of wives, without having to arrange them in a series...
eBook - ePubDeleuze and the History of Mathematics
In Defense of the 'New'
- Simon Duffy(Author)
- 2013(Publication Date)
- Bloomsbury Academic(Publisher)
...The analytical expressions involving numbers and letters, rather than the geometric objects for which they stood, became the focus of interest. It was this change of focus toward the formula that made the emergence of the concept of function possible. In this process, the differential underwent a corresponding change; it lost its initial geometric connotations and came to be treated as a concept connected with formulas rather than with figures. With the emergence of the concept of the function, the differential was replaced by the derivative, which is the expression of the differential relation as a function, first developed in the work of Euler (b. 1707–1783). One significant difference, reflecting the transition from a geometric analysis to an analysis of functions and formulas, is that the infinitesimal sequences are no longer induced by an infinitangular polygon standing for a curve, according to the law of Continuity as reflected in the infinitesimal calculus, but by a function, defined as a set of ordered pairs of real numbers. Subsequent developments in mathematics: The problem of rigor The concept of the function, however, did not immediately resolve the problem of rigor in the calculus. It was not until the late nineteenth century that an adequate solution to this problem was posed. It was Karl Weierstrass (b. 1815–1897) who “developed a pure nongeometric arithmetization for Newtonian calculus” (Lakoff and Núñez 2000, 230), which provided the rigor that had been lacking. The Weierstrassian program determined that the fate of calculus need not be tied to infinitesimals, and could rather be given a rigorous status from the point of view of finite representations...
