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Question

Prove that greatest integer function is continuous at all the points except at integer points.

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Solution

Checking continuity when x=2
(i) f(x)=x for all x ε R
By definition of greatest integer function, if x lies between two successive integers then f(x)=least integer of them.
(ii) So, at x=2, f(x)=[2]=2 --------(1)
left side limit
(x2-h):f(x)=[2-h]=1 -------(2)
[Since (2-h) lies between 1 and 2 and the least being 1]
right side limit
(x2+h):f(x)=[2+h]=1 -------(3)
[Since (2+h) lies between 2 and 3 and the least being 2]
(iii) Thus from above 3 equations left side limit is not equal to right side limit.
So, limit of function does not exist.
Hence, it is discontinuous at x=2.
So, greatest integer function is not constant at all points.

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