Let f: R R be defined as f(x) = cases a e^x - b, & x 0 b + [x^2 - x], & 0 where a, b, c R , [t] denotes the greatest integer less than or equal to t , t denotes the fractional part of t , and sgn (t) denotes the signum function. If f is discontinuous at exactly one point, then which of the following statements is true?

Step-by-step solution

For f(x) to be continuous at x=0 : _ x 0^- f(x) = f(0) = a e^0 - b = a - b For 0 _ x 0^+ f(x) = _ x 0^+ (b - 1) = b - 1 Thus, a - b = b - 1 a = 2b - 1 At x=1 : _ x 1^- f(x) = b - 1 f(1) = c + 1 = c + 0 = c _ x 1^+ f(x) = _ x 1^+ (c + x - 1) = c For continuity at x=1 , b - 1 = c b = c + 1 At x=2 : _ x 2^- f(x) = _ x 2^- (c + x - 1) = c + 1 f(2) = [2] + sgn (0) = 2 + 0 = 2 _ x 2^+ f(x) = _ x 2^+ ([x] + sgn (x - 2)) = 2 + 1 = 3 Since f(2) _ x 2^+ f(x) , f(x) is discontinuous at x=2 for all values of a, b, c . Since f(