LimitsHard
Question
If f(x) = 
, where { . } denotes the fractional part function, then:
, where { . } 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) = 
f(x) = 0 - 1 + 0.sin(-1) = - 1
f(0+) =
f(x) = 0 + 0 + 0.sin0 = 0 f(0)
f(x) is not continuous at x = 0
at x = 2,
f(2+) = 2 + 2 + 2 sin 2 = 4 + 2 sin 2
f(2-) = 2 + 1 + 2 sin 1 = 3 + 2 sin 1
f(x) is not continuous at x = 2
f(x) = 0 - 1 + 0.sin(-1) = - 1f(0+) =
f(x) = 0 + 0 + 0.sin0 = 0 f(0)f(x) is not continuous at x = 0
at x = 2,
f(2+) = 2 + 2 + 2 sin 2 = 4 + 2 sin 2
f(2-) = 2 + 1 + 2 sin 1 = 3 + 2 sin 1
f(x) is not continuous at x = 2
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