Continuity and DifferentiabilityHard
Question
If f(x) = 
where {x} denotes the fractional part function, then -
where {x} denotes the fractional part function, then -Options
A.′f′ is continuous & differentiable at x = 0
B.′f′ is continuous but not differentiable at x = 0
C.′f′ is continuous & differentiable at x = 2
D.none of these
Solution
f′(0+) = 
=
= 2
f′(0-) =
=
=
- 2 + 
- sin (1 - h)
⇒ LHD does not exist
hence function is non differentiable and discontinuous at x = 0. Similarly for x = 2.
=
= 2f′(0-) =
=
=
- 2 + - sin (1 - h)⇒ LHD does not exist
hence function is non differentiable and discontinuous at x = 0. Similarly for x = 2.
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