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find all the points of discontinuity of the function f(x) defined by {:{(x+1, if x lt 1), (1 , if x =1), (x-1, if x gt1):} |
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Answer» Solution :The function f is defined at all points of the real line. Case I : Let x=c be any ARBITRARY point in the domain of f(x) ifc lt 1 , f(c)=c+1 , therefore `underset(x to c) lim f(x) = underset(x to c) lim (x+1) = c+1` Thus, f is continuous at all real numbers less than 1. Case II : If cgt 1 , then f(c) = c -1 , therefore `underset(x to c)lim f(x)= underset(x to c) lim (x -1) = c-1 = f(c)` Thus f is continuous at all point x gt 1 Case III. C =1 , the left hand LIMIT of f(x) at x =1 is ` underset(x to 1^(-)) lim f(x) = underset(x to 1^(-)) lim (x +1) = 1 + 1 = 2 ` The right hand limit of f at x =1 is ` underset(x to 1^(+)) lim f(x) = underset(x to 1^(+)) lim (x-1) = (1-1) =0` Since, the left and right hand limits of f at x =1 do not coincide, f is not continuous at x =1, Hence, x =1 is the only point of DISCONTINUITY of f. The graph of the function is as SHOWN. 
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