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Question
The function f (x) = x − [x], where [⋅] denotes the greatest integer function is
Options
continuous everywhere
continuous at integer points only
continuous at non-integer points only
differentiable everywhere
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Solution
(c) continuous at non-integer points only
\[\text{ We have }, \]
\[f\left( x \right) = x - \left[ x \right]\]
\[\text{Consider n be an integer} . \]
`f(x) = x - [x] = {(x-(n-1),n-1le x <n),(0, x = n),(x-n , n le x < n +1):}`
Now,
\[\left( \text { LHL at x = n } \right) = \lim_{x \to n^-} f\left( x \right) = x - \left( n - 1 \right) = x - n + 1\]
\[\left( \text { RHL at x } = n \right) = \lim_{x \to n^+} f\left( x \right) = x - \left( n \right) = x - n\]
\[\text { As, LHL } \neq\text { RHL at x } = n\]
\[i . e . , \text{given function is not continuous at n} . \]
\[\text{Now, n is any integer} . \]
\[\text{Therefore, given function is not continuous at integers} . \]
Therefore, given points are continuous at non-integer points only.
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