KVPY2017MathematicsContinuity and Differentiability
Define g ( x )= _ -3 ³ f ( x - y ) f ( y ) dy , for all real x , where f(t)= array ll 1, & 0 t 1 0, & elsewhere array . Then
Options
- Ag ( x ) is not continuous everywhere
- Bg ( x ) is continuous everywhere but differentiable nowhere
- Cg ( x ) is continuous everywhere and differentiable everywhere except at x =0,1
- Dg(x) is continuous everywhere and differentiable everywhere except at x =0,1,2
Correct answer
D. g(x) is continuous everywhere and differentiable everywhere except at x =0,1,2
Step-by-step solution
Definition can be break as array l g(x)= ₀¹ f(x-y) d y x-y=t ;-d y d t g(x)= _ x-1 ^ x f(t) d t g(x)= array cc 0 & x 0 x & 0 2 array . array Now, check yourself