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Question
Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.
Solution
Any function will not be differentiable if the left hand limit and the right hand limit are not equal.
f(x) = [x], 0 < x < 3
(i) At x = 1
Left side limit = `lim_(h -> 0) ([1 - h] - [1])/-h`
= `lim_(h -> 0) (0 - 1)/-h`
= `lim_(h -> 0) 1/h`
= infinite
right hand limit
= `lim_(h -> 0) ([1 + h] - [1])/h`
= `lim_(h -> 0) (1 - 1)/h`
= 0
Left side limit and right side limit are not equal.
Hence, f(x) is not differentiable at x = 1.
(ii) At x = 2
left side limit
= `lim_(h -> 0) (f(2 + h) - f(2))/h`
= `lim_(h -> 0) ([2 + h]-2)/h`
= `lim_(h -> 0) (2 -2)/h`
= 0
right hand limit
= `lim_(h -> 0) (f(2 - h) - f (2))/h`
= `lim_(h -> 0) ([2 - h] - [2])/-h`
= `lim_(h -> 0) (1 - 2)/-h`
= infinite
Left side limit and right side limit are not equal.
Hence, f(x) is not differentiable at x = 2.
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