Let \([t]\) denote the greatest integer \(\le t\) and \(aZ = \{ax : x \in Z, a \in R\}\) (where Z is set of integer and R is set of real number). The set of points of discontinuity of the function \(f(x) = [2x]\) is given by
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{1}{2}Z\)
The greatest integer function [t] is continuous at every non-integer t and jumps by 1 at every integer t. So \(f(x) = [2x]\) is discontinuous exactly where 2x is an integer:
\(\displaystyle 2x = n,\; n \in Z \Rightarrow x = \frac{n}{2}\)
For example, at \(x = \frac{1}{2}\): just to the left, 2x is slightly less than 1, so f(x) = 0, while \(f\left(\frac{1}{2}\right) = 1\). So f is discontinuous there, although \(\frac{1}{2}\) is not an integer.
The set of points of discontinuity is
\(\displaystyle \left\{\frac{n}{2} : n \in Z\right\} = \left\{\ldots, -1, -\frac{1}{2}, 0, \frac{1}{2}, 1, \ldots\right\} = \frac{1}{2}Z\)