Continuity expresses the property of being uninterrupted. Intuitively, a continuous line or function is one that can be graphed without having to lift the pencil from the paper; there are no missing points, no skipped segments and no disconnections. This intuitive notion of continuity goes back to ancient Greece, where many mathematicians and philosophers believed that reality was a reflection of number. Thus, they thought, since numbers are infinitely divisible, space and time must also be infinitely divisible. In the fifth century B.C., however, the Greek mathematician Zeno pointed out that a number of logical inconsistencies arise when assuming that space is infinitely divisible, and stated his findings in the form of paradoxes. For example, in one paradox Zeno argued that the infinite divisibility of space actually meant that all motion was impossible. His argument went approximately as follows: before reaching any destination a traveler must first complete one-half of his journey, and before completing one-half he must complete one-fourth, and before completing one-fourth he must complete one-eighth, and so on indefinitely. Any trip requires an infinite number of steps, so ultimately, Zeno argued, no journey could ever begin, and all motion was impossible. Zeno's paradoxes had a disturbing effect on Greek mathematicians, and the ultimate resolution of his paradoxes did not occur until the intuitive notion of continuity was finally dealt with logically.
The continuity of space or time, considered by Zeno and others, is represented in mathematics by the continuity of points on a line. As late as the seventeenth century, mathematicians continued to believe, as the ancient Greeks had, that this continuity of points was a simple result of density, meaning that between any two points, no matter how close together, there is always another. This is true, for example, of the rational numbers. However, the rational numbers do not form a continuum, since irrational numbers like √ 2 are missing, leaving holes or discontinuities. The irrational numbers are required to complete the continuum. Together, the rational and irrational numbers do form a continuous set, the set of real numbers. Thus, the continuity of points on a line is ultimately linked to the continuity of the set of real numbers, by establishing a one-to-one correspondence between the two. This approach to continuity was first established in the 1820s, by Augustin-Louis Cauchy, who finally began to solve the problem of handling continuity logically. In Cauchy's view, any line corresponding to the graph of a function is continuous at a point, if the value of the function at x, denoted by f(x), gets arbitrarily close to f(p), when x gets close to a real number p. If f(x) is continuous for all real numbers x contained in a finite interval, then the function is continuous in that interval. If f(x) is continuous for every real number x, then the function is continuous everywhere.
Cauchy's definition of continuity is essentially the one we use today, though somewhat more refined versions were developed in the 1850s, and later in the nineteenth century. For example, the concept of continuity is often described in relation to limits. The condition for a function to be continuous, is equivalent to the requirement that the limit of the function at the point p be equal to f(p), that is:
In this version, there are two conditions that must be met for a function to be continuous at a point. First, the limit must exist at the point in question, and, second, it must be numerically equal to the value of the function at that point. For instance, polynomial functions are continuous everywhere, because the value of the function f(x) approaches f(p) smoothly, as x gets close to p, for all values of p.
However, a polynomial function with a single point redefined is not continuous at the point x = p if the limit of the function as x approaches p is L, and not f(p). This is a somewhat artificial example, but it makes the point that when the limit of f(x) as x approaches p is not f(p) then the function is not continuous at x = p. More realistic examples of discontinuous functions include the square wave, which illustrates the existence of right and left hand limits that differ; and functions with infinite discontinuities, that is, with limits that do not exist.
These examples serve to illustrate the close connection between the limiting value of a function at a point, and continuity at a point.
There are two important properties of continuous functions. First, if a function is continuous in a closed interval, then the function has a maximum value and a minimum value in that interval. Since continuity implies that f(x) cannot be infinite for any x in the interval, the function must have both a maximum and a minimum value, though the two values may be equal. Second, the fact that there can be no holes in a continuous curve implies that a function, continuous on a closed interval [a,b], takes on every value between f(a) and f(b) at least once. The concept of continuity is central to isolating points for which the derivative of a function does not exist. The derivative of a function is equal to the slope of the tangent to the graph of the function. For some functions it is not possible to draw a unique tangent at a particular point on the graph, such as any endpoint of a step function segment. When this is the case, it is not possible to determine the value of the derivative at that point. Today, the meaning of continuity is settled within the mathematics community, though it continues to present problems for philosophers and physicists.
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