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Disucss the continuity of the function f given by f (x) = {:{ (-x, if x ge0) , (-x^(2) ,if x lt 0):} |
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Answer» Solution :The function is defined at every REAL number. Graph is as shown. The domain of definition of f(x) inotthree DISJOINT subsets of the real line. Let ` D_(1) ={ x in R: x lt 0) ` ` D_(2) = {0}` `D_(3) = {x in R : x gt 0}` CASE I : x lt 0 , ` f(x) = -x^(2)` the function is continuous at all x lt 0 Case II : x gt 0 , f(x) = -x the function is continuous at all x gt 0 Case III : x=0 , f(0)=0 The left hand LIMIT of f(x) at x =0 is ` underset(x to 0^(-))lim f(x) = underset(x to 0^(-)) lim (-x) =0 ` The right hand limit of f(x) at x = 0 is ` underset(x to x^(+)) lim f(x) = lim(x to 0^(+)) ( -x) =0` The ` underset(x to 0) f(x) = 0= f(0) ` and HENCE f(x) is continuous at x =0 . This means that f(x) is continuous at every point in its domain. Hence , f(x) is a continuous function. 
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