KVPY2019MathematicsContinuity and Differentiability
Let f : R → R be a function defined by f x = sin x 2 x   if   x ≠ 0 0   if   x = 0 Then, at x = 0 , f is
Options
- Anot continuous
- Bcontinuous but not differentiable
- Cdifferentiable and the derivative is not continuous
- Ddifferentiable and the derivative is continuous
Correct answer
D. differentiable and the derivative is continuous
Step-by-step solution
Given function f x = sin x 2 x ,   x ≠ 0 0 ,   if   x = 0 then lim x → 0 f x = lim x → 0 sin x 2 x = lim x → 0 x sin x 2 x 2 = 0 = f 0 Hence, f x is continuous at x = 0 Now, for differentiability RHD (at x = 0 ) = lim h → 0 f 0 + h - f 0 h = lim h → 0 sin h 2 h 2 = 1 and LHD (at x = 0 ) = lim h → 0 f 0 - h - f 0 - h = lim h → 0 sin h 2 h 2 = 1 So, f x is differentiable at x = 0 ∴   f ' x = 2 cos x 2 - sin x 2 x 2 ,   if   x X