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Question

Let f(x)=[x]+[x]. Then for any integer n and kRI

A
limxnf(x) exists
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B
limxkf(x) exists
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C
f is continuous at x=n
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D
f is continuous at x=k
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Solution

The correct options are
A limxnf(x) exists
B limxkf(x) exists
D f is continuous at x=k
f(n)=[n]+[n]=nn=0
limxnf(x)=limh0+[nh]=[n+h]=n1+(n)=1
limxn+f(x)=limh0+[n+h]=[nh]=n+(n1)=1
If kI, the function [x] and [x] are continuous at k.
Hence limxk f(x) exists.

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