. Differential and integral calculus, an introductory course for colleges and engineering schools. Eig.(c) Fig.(d) discontinuities can pass from one side of the #-axis to the otherside only as in (a), by actually crossing that axis. Such a curve isthe graph of a real single-valued and continuous function. Wehave, then, Theorem 1. A real, single-valued, and continuous function whichhas both + and — values within an interval must be zero at least oncewithin that interval. Or, A real, single-valued, and continuous function can change sign onlyby passing through zero. The following theorem follows
. Differential and integral calculus, an introductory course for colleges and engineering schools. Eig.(c) Fig.(d) discontinuities can pass from one side of the #-axis to the otherside only as in (a), by actually crossing that axis. Such a curve isthe graph of a real single-valued and continuous function. Wehave, then, Theorem 1. A real, single-valued, and continuous function whichhas both + and — values within an interval must be zero at least oncewithin that interval. Or, A real, single-valued, and continuous function can change sign onlyby passing through zero. The following theorem follows directly from theorem 1: its truthbecomes apparent on drawing a figure. Theorem 1. If fix) is real, single-valued, and continuous through- fun like)like \ siyns> then between a and b f(x) is zero an \ > number of times: the J K J (even ) even number may of course be zero. §51 SOME GENERAL PROPERTIES OF FUNCTIONS 65 The converse of this theorem is not universally true. That is,iff(x) = 0 within an interval, we cannot assert that f(x) has both YY.
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