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BWPLE2 Show that fx) Ix] is not continuous at x -n, where n is an in71 15 011 |
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Answer» i) f(x) = [x], for all x in R ==> By the definition of greatest integer function: If x lies between two successive integers, then f(x) = least integer of them. ii) So, at x = 2, f(x) = [2] = 2 -------- (1) Left side limit (x ---> 2-h): f(x) = [2 - h] = 1 ----- (2) {Since (2 - h) lies between 1 & 2; and the least being 1} Right side limit (x --> 2+h): f(x) = [2 + h] = 2 -------- (3) {Since (2+h) lies between 2 & 3; and the least being 2} iii) Thus from the above 3 equations, left side limit is not equal to right side limit. So limit of the function does not exist. Hence it is discontinuous at x = 2 So this is not derivable at x = 2 Hence Proved. |
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