Let a function f : R → R be defined as, f x = sin x − e x if x ≤ 0 a + - x if 0 < x < 1 2 x − b if x ≥ 1 Where x is the greatest integer less than or equal to x . If f is continuous on R , then a + b is equal to:

Step-by-step solution

We have, f x = sin x − e x   if     x ≤ 0 a + - x   if     0 < x < 1 2 x − b   if     x ≥ 1 If f x is continuous at x = a then lim x → a - f x = f a = lim x → a + f x . Continuous at x = 0 f 0 + = f 0 - ⇒   a - 1 = 0 - e 0 ⇒   a = 0 Continuous at x = 1 f 1 + = f 1 - ⇒     2 1 - b = a + - 1 ⇒     b = 2 - a + 1 ⇒   b = 3 ∴     a + b = 3