JEE Main202125 Jul 2021Morning ShiftMathematicsContinuity and DifferentiabilityActual
Let f : [ 0 , ∞ ) → [ 0 , ∞ ) be defined as f x = ∫ 0 x y d y where [ x ] is the greatest integer less than or equal to x . Which of the following is true?
Options
- Af is continuous at every point in [ 0 , ∞ ) and differentiable except at the integer points.
- Bf is both continuous and differentiable except at the integer points in [ 0 , ∞ ) .
- Cf is continuous everywhere except at the integer points in [ 0 , ∞ ) .
- Df is differentiable at every point in [ 0 , ∞ ) .
Correct answer
A. f is continuous at every point in [ 0 , ∞ ) and differentiable except at the integer points.
Step-by-step solution
f : [ 0 , ∞ ) → [ 0 , ∞ ) , f x = ∫ 0 x y d y Let x = n + f , f ∈ ( 0 , 1 ) ⇒ x = n ,   x = f . So f x = ∫ 0 1 y d y + ∫ 1 2 y d y + ∫ 2 3 y d y + … + ∫ n - 1 n y d y + ∫ n n + f y d y f x = ∫ 0 1 0 d y + ∫ 1 2 1 d y + ∫ 2 3 2 d y + … + ∫ n - 1 n n - 1 d y + ∫ n n + f n d y f x = 0 1 - 0 + 1 2 - 1 + 2 3 - 2 + … + n - 1 n - n - 1 + n n + f - n f x = 0 + 1 + 2 + … + n - 1 + n f ⇒ f x = n