. Differential and integral calculus, an introductory course for colleges and engineering schools. unction has no value whatever, in other words, does not f(x) has no value when x = a, that is, if f(a) does not exist,f(x) is said to be discontinuous when x = a, or to have a discon-tinuity when x = a. Its graph will lack the point whose abscissais a: there will be at that point a break in the continuity of thecurve. The following are examples of functions with discontinu-ities. Bin T (1) The function has no existence when x = 0. For when x x = 0, the function takes the form -, which ha
. Differential and integral calculus, an introductory course for colleges and engineering schools. unction has no value whatever, in other words, does not f(x) has no value when x = a, that is, if f(a) does not exist,f(x) is said to be discontinuous when x = a, or to have a discon-tinuity when x = a. Its graph will lack the point whose abscissais a: there will be at that point a break in the continuity of thecurve. The following are examples of functions with discontinu-ities. Bin T (1) The function has no existence when x = 0. For when x x = 0, the function takes the form -, which has no value because0 cannot be used as a divisor. The function has, therefore, a dis-continuity when x = 0. For all other finite values of x, has INTRODUCTION 11 a definite and determinate value. A clear insight into the natureof this singularity may be had by a study of the graph of the func-tion as shown in the figure. Its equation is of course y = sinz x For very small + and — values of x, sin x x differs little from 1, and the smaller the numerical value of x, the nearer does the function. come to the value 1, so that the curve approaches as near as weplease to the point (0, 1) from either side of the y-axis. The point(0, 1), however, does not itself belong to the graph: the graphmay be regarded as broken in two at this point or as having had ahole punched in it here. This point is a singular point, or a pointof discontinuity of the graph, or of the function. (2) The function — has no existence when x = 0. For when x = 0, the function has the form - , which has no value because it is impossible to divide 1 by function has, therefore, adiscontinuity when x = 0, buthas a determinate value for allother values of x. The graphis shown in the figure. It hasno point whose abscissa is 0,that is, it has no point in com-mon with the 2/-axis. When x
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