The function f (x) = x − [x], where [⋅] denotes the greatest integer function is

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(c) continuous at non-integer points only 	ext{ We have }, f\left( x ight) = x - \left[ x ight] 	ext{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( 	ext { LHL at x = n } ight) = \lim_{x 	o n^-} f\left( x ight) = x - \left( n - 1 ight) = x - n + 1 \left( 	ext { RHL at x } = n ight) = \lim_{x 	o n^+} f\left( x ight) = x - \left( n ight) = x - n 	ext { As, LHL } 
eq	ext { RHL at x } = n i . e . , 	ext{given function is not continuous at n} . 	ext{Now, n is any integer} . 	ext{Therefore, given function is not continuous at integers} . Therefore, given points are continuous at non-integer points only.

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