. Differential and integral calculus, an introductory course for colleges and engineering schools. the figure,we may speak of the behaviorof the function along the arcAB instead of within the in-terval a ^ x = b. The intervalmay be very small, or it may be large enough to contain the entire graph. We may even have the interval — oo = x = + oo . 48. Continuity. Confining ourselves to real values of functions(it is only for real values that a function has a graph), we define afunction to be continuous within an interval when its graph is un-broken within that interval, and to have a discontinuit
. Differential and integral calculus, an introductory course for colleges and engineering schools. the figure,we may speak of the behaviorof the function along the arcAB instead of within the in-terval a ^ x = b. The intervalmay be very small, or it may be large enough to contain the entire graph. We may even have the interval — oo = x = + oo . 48. Continuity. Confining ourselves to real values of functions(it is only for real values that a function has a graph), we define afunction to be continuous within an interval when its graph is un-broken within that interval, and to have a discontinuity whereverthere is a break in its graph* In Art. 8 we virtually defined a discontinuity as a point at which * Note to the Teacher. — This geometrical definition and discussion of conti-nuity is lacking in rigor, in that it appeals to the geometrical intuition, but itsuffices for the purposes of an elementary course, and wall be found to appealmore strongly to the beginning student than would a more rigorous rigorous analytical definition is given at the end of this article. 60. §48 SOME GENERAL PROPERTIES OF FUNCTIONS 61 the function becomes nonexistent. At such a point there is abreak in the graph and the discontinuity falls under our presentdefinition, but we shall now give a couple of examples showingthat a break in the graph may occur even when the function doesnot cease to exist there, and that consequently our present defini-tion is more comprehensive than the former one. Example 1. Let the function G(x) mean the greatest whole numberin x so that when -1< x < 1, G(x) = 0; when l^x < 2, G(x) = 1; when2 ^ x < 3, G(x) = 2; . . when- 2 < x ^ -1, G(x) = -1; and ingeneral n being a positive integer,when n^x<n-\-l, G(x) = n, andwhen — (n+ 1) < x= - n, G(x) =-n. The graph of G{x) consistsof a series of discrete line seg-ments parallel to OX, and oflength 1. The function has dis-continuities, breaks in its graph,at the points x = ±l, ±2,
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