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                                    Discuss the continuity of the function f defined byf(x)= {:{(x +3, if x le 1 ), (x -3, if x gt 1):} | 
                            
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Answer» Solution :The function f(x) is definedat all points of the real LINE.  Case I : LET x = c be any arbitrary real number in the domain of f(x). If c lt 1 , then f(c ) = c +3 , therefore ` underset(x to c) lim f(x) = underset(x to c)lim (x+3) = c +3` Thus, f is CONTINUOUS at all real numbers less than 1. Case II. If c gt 1then f(c) = c -3 , therefore ` underset(x to c) lim f(x) = underset(x to c) lim (x-3) = (c -3) = f(c)` Thus, f(x) is continuous at all points x gt 1 . Case III. If c=1,then the left hand limit of f(c) at x =1 ` underset(x to 1^(-))lim f(x) = underset(x to 1^(+))(x -3) = 1 -3 = -2` Since the left and right hand LIMITS of f(x) at x=1do not coincide, f(x)is , not continuous at x =1 . Hence x =1 is the only pointof DISCONTINUITY of f(x). the graph of the function is as show.  
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