Continuity at a point.Let a be a real numbers. Suppose f(x) is a real-valued function for which
for some number L (finite). Moreover, suppose f(a) is defined and L = f(a). then f(x) is continuous at x = a. The following theorem will help you quickly identify continuous functions in any calculus course that you take. Theorem: Continuity at a Point.
In practice, continuity at x = a implies the limit
Example:
Continuity of a function f(x) at a number a corresponds to the requirement that the limit L = f(a). That is, f(a) = limx® a f(x) In the latter case, limit evaluation by immediate substitution is possible. It is possible for
In the latter case f(x) is not continuous at x = a. Continuity on an Interval.A function f(x) is continuous on an interval I if and only if for each point a in I, the function f(x) is continuous at x = a. (Here is understood that one sided limits are to be used at included endpoints of the interval I.) Theorem: Continuity on an Interval.
Continuity at Point RevisitedTo explain the idea of continuity of a function y = f(x) at a point x = a, we ask the following error-control question with b = f(a): to what number m of places should the decimal expansions of x and a agree, for the decimal expansion of the number f(x) to agree with that of b = f(a) to n-decimal places? That is, given a whole number n, is there an m such that
The Greek letters d (delta) and e (epsilon) above are employed here in accordance with tradition of some (not all) calculus texts. For simplicity, the error control tolerances e and d in the first instance here and below, may be restricted to be numbers of the form ½ ·10-k = ½ [1/(10k)]. The decimal free discussion of error control and continuity dispenses with this requirement. We say a function f(x) is continuous at a point x = a if and only if unlimited error control is possible there. More formally, we state the following definition. Theorem 14.1 [Continuity at a Point] If f(x) is a real-valued function of a real number x in an interval [c,d], and a is a number in the interval [c,d] then the function f is said to be continuous at the number x = a if and only if the following holds. If for every n, there exist an m such that
Decimal-Free FormThe decimal-free description or definition of continuity at a point x = a is as follows.[Continuity at Point] If f(x) is a real-valued function of a real number x in an interval [c,d], and a is a point in the interval [c,d] then the function f is said to be continuous at x = a if and only if the following holds: For every e1 > 0, there exist a d1 > 0 such that
Proof of Equivalence. To show the decimal-based description implies the decimal-free description of continuity, observe the following. First given e1 > 0, there is an n > 0 such that e1 > ½·[1/(10n)] = e. The decimal-based requirement for continuity now is satisfied for some d = ½·[1/(10m)]. So the decimal-free version holds with d1 = d = ½·[1/(10m)]. Conversely, the other way that is, to show the latter decimal-free form implies the decimal-based description of continuity, observe the following. Given m > 0, let e1 = e = ½ ·[1/(10m)]. Then choose d1 > 0 so that the decimal-free requirement is satisfied. The decimal-based version is then satisfied if m > 0 is selected so that d1 ³ d = ½·[1/(10m)]. Recapitulation - Limit of a Function
Suppose f(x) is a function of real numbers x and that it is defined on an interval containing the number a. [Limit of a Function] A function f(x) converges to a finite limit at the point x = a if and only if there is a real number L such that for every integer n, there is an m such that
Continuity of a function f(x) at a number a corresponds to the requirement that the limit L = f(a). But it is possible for the limit L = limx® a f(x) to exist and not equal f(a). Significant Digit Error Control
The question of relative error is related to the unrestricted control of the number of significant digits in computations: For every n is there an m such that
This question can only be answered when division by zero is avoided. In numerical calculations, circumstances may suggest what is more important (more precisely what is feasible): absolute error control or relative error control. Various error control (or continuity) questions can be based on different
measures of closeness for x and f(x), that is, different
measures of closeness on the domain and range of a function f. For
example, the question of relative error on the domain can also be posed as
follows: for every n is there an m such that
For addition and subtraction, absolute error control (the first type introduced in this chapter) is more appropriate than relative error or significant digit control. For multiplication and division, relative error and significant digit error control is more appropriate. When there is a mixture of addition or subtraction with multiplication or division, no simple advice can be offered. A course on numerical methods could discuss this topic further.
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