. Differential and integral calculus, an introductory course for colleges and engineering schools. uous merely by taking complex values. For example,Vx2 — 1 is imaginary for all values of x between —1 and +1, butis nowhere nonexistent or discontinuous. The rigorous analytical definition of continuity is as follows:If f(a) exists, and if lim f{x) = f(a), no matter in what way x x=a approaches its limit a, f(x) is defined to be continuous at a. If, on the other hand, f(a) does not exist, or if, when f(d) does existx can be made to approach a in such a way that lim f(x) y£ f{a), f(x) x=a is defin
. Differential and integral calculus, an introductory course for colleges and engineering schools. uous merely by taking complex values. For example,Vx2 — 1 is imaginary for all values of x between —1 and +1, butis nowhere nonexistent or discontinuous. The rigorous analytical definition of continuity is as follows:If f(a) exists, and if lim f{x) = f(a), no matter in what way x x=a approaches its limit a, f(x) is defined to be continuous at a. If, on the other hand, f(a) does not exist, or if, when f(d) does existx can be made to approach a in such a way that lim f(x) y£ f{a), f(x) x=a is defined to be discontinuous at a. It can be shown that every case of discontinuity thus far men-tioned is covered by this definition. §§49-50 SOME GENERAL PROPERTIES OF FUNCTIONS 63 49. Discontinuities of f(x). A function may itself be con-tinuous within an interval and yet have a derivative which isdiscontinuous within that interval. Such are the functions whosegraphs are shown in the accompanying figures. In the first figurethe derivative is discontinuous at A, B, and C, because at each of.
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