For real x with -10 x 10 define f(x)= _ -10 ^ x 2^ [t] d t , where for a real number r we denote by [r] the largest integer less than or equal to r. The number of points of discontinutiy of f in the interval (-10,10) is

Step-by-step solution

Let r be an integer in (-10,10) array l Now, L H L= _ x r⁻ _ -10 ^ x 2^ [t] d t = _ h 0⁺ [ _ -10 ⁻⁹ 2^ [t] d t+ _ -9 ⁻⁸ 2^ [t] d t+ + _ r-1 ^ r-h 2^ [t] d t ] = _ h 0 [2⁻¹⁰+2⁻⁹+ . .+2^ r-1 (1-h) ] =2⁻¹⁰+2^ - + +2^ n-1 ..(1) array array l _ x r⁺ _ -10 ^ x 2^ [t] d t = _ h 0⁺ [ _ -10² ⁻⁹ 2^ [t] d t+ _ -9 ⁻⁸ 2^ [t] d t+ + _ r ^ r+h 2^ [t] d t ] =2⁻¹⁰+2⁻⁹+ . .+2^ n-1 ...(2) f(r)= _ -10 ^ r 2^ [t] d t =2⁻¹⁰+2⁻⁹+ +2⁻¹..(3) array From (1),(2) &(3)f(x) is continous-u : integers. f(x) is continous at all integers.