InterviewSolution
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InterviewSolution
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Find all the points of discontinuity of the greatest integer function defined by f(x) = [x] , where [x]denotes the greatest integer less than or equal to x. |
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Answer» Solution :Graph of the function is given in FIGURE. From the graph, it seems like that f(x) is dicontinuousat every integral point of x. Case I : Let c be a real number which , is not equal to any integer. It is evident from the graph that for all real numbers close to c the value of the function is equal to [c] , i.e,` underset (x to c) lim f(x) = underset(x to c) lim [x] = [c] ` , Also f (c) = [c] and hence the function is continuous at all real number not equal to integers. Case II : Let c be an integer. Then we can find a SUFFICIENTLY small real number h GT 0 i.e.0 LT h lt 1. such that [c -h] = c-1 WHEREAS =[c + h] =c Thus, ` underset( x to c^(-)) lim f(x) = c -1 and underset(x to c^(+)) f (x) =c ` Since these cannotbe equal to each other for any c, the function is discontinuous at every integral point. 
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