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Question

Show that the function defined by g(x)=x[x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.

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Solution

The given function g(x) is defined at all integral points.

Let n be an integer.

Then g(n)=nn=nn=0

L.H.L. at x=n=limxng(x)=limxn(x[x])=n(n1)=1

R.H.L. at x=n=limxn+g(x)=limxn+(x[x])=nn=0

Since L.H.L.R.H.L.

Therefore g is not continuous at x=n. i.e., g(x) is discontinuous at all integral points.

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