Most Important Selected Qs for JEE AdvancedMathematicsContinuity and Differentiability

Consider the function f(x)= array ll x, 1 x x, 1 x , & when x 0 1, & when x=0 array . , then:

Options

A_ x 0⁺ f(x) 0
B_ x 0⁻ f(x)=0
Cf(x) is continuous for all x 0
Df(x) is derivable for all x 0

Correct answer

C. f(x) is continuous for all x 0

Step-by-step solution

x, 1 x = cases x, & x [1, ) 1 x , & x (0,1) x, & x [-1,0) 1 x , & x (- ,-1) cases and . x, 1 x = cases 1 x , & x [1, ) x, & x (0,1) 1 x , & x [-1,0) x, & x (- ,-1) cases f(x)= cases x^2, & x 1 1 x^2 , & 0 < x < 1 x^2, & -1 x < 0 1 x^2 , & x < -1 1, & x-0 cases _ x 0⁺ f(x) 0, _ x 0⁻ f(x)=0 f(x) is discontinuous at x=0 .

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