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Question
Show that the function g (x) = x − [x] is discontinuous at all integral points. Here [x] denotes the greatest integer function.
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Solution
Given: `g(x)=x-[x]`
It is evident that g is defined at all integral points.
Let \[n \in Z\] .
Then,
`g(n)=n-[n]=n-n=0`
The left hand limit of f at x = n is,
`lim_(x->n^-)g(x)=lim_(x->n^_)(x-[x])=lim_(x->n^_)(x) -lim_(x->n^_)[x]=n-(n-1)=1`
The right hand limit of f at x = n is,
`lim_(x->n^+)g(x)=lim_(x->n^+)(x-[x])=lim_(x->n^+)(x)-lim_(x->n^+)[x]=n-n=0`
It is observed that the left and right hand limits of f at x = n do not coincide.
i.e.
So, f is not continuous at x = n,
Hence, g is discontinuous at all integral points.
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