The function f(x)=(x) where (x) denotes the smallest integer `ge x` isA. everywhere continuousB. continuous at x=n, `n in Z`C. continuous on R-ZD. none of these

The function f(x)=(x) where (x) denotes the smallest integer `ge x` is

1.	The function f(x)=(x) where (x) denotes the smallest integer `ge x` isA. everywhere continuousB. continuous at x=n, `n in Z`C. continuous on R-ZD. none of these
Answer» Correct Answer - C For any `n in Z`, we have `("LHL at x=n")=underse(x to n^(-))lim f(x)` `Rightarrow ("LHL at x=n")=underse(h to 0)lim f(n-h)=underset(h to 0)lim (n-h)=n` `("RHL at x=n")=underset(x to n^(+))lim f(x)` `Rightarrow ("RHL at x=n")=underset(h to 0)lim f(n+h)=n+1` `therefore underset(x to n^(-))lim f(x)ne underset(x to n^(+))lim f(x)` Let `x=a in R-Z`. Then, there exists `n in Z` such that `n lt a lt n+1` Now, `underset(x to a^(-))lim f(x)=underset(x to 0)limf(a-h)=underset(h to 0)lim (a-h)=n+1` `underset(x to a^(+))lim f(x)=underset(h to 0)limf(a+h)=underset(h to 0)lim (a+h)=n+1` `and, f(a)=n+1` `therefore underset(x to a^(-))lim f(x)=underset(x to a^(+))lim f(x)=f(a)` There f(x) is a continuous at x=a Hence, f(x) is a continous at all points other than integer points.