Let $[t]$ denote the greatest integer $≤t$ and $aZ = \{ax: x ∈ Z,a ∈ R\}$ (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

$\frac{1}{2}Z$

The correct answer is Option (3) → $\frac{1}{2}Z$

Given function: $f(x) = [2x]$, where $[t]$ denotes the greatest integer less than or equal to $t$.

The greatest integer function $[t]$ is discontinuous at all integer values of $t$.

Hence, $f(x) = [2x]$ will be discontinuous when the argument $2x$ is an integer.

That is,

$2x = n \;\Rightarrow\; x = \frac{n}{2}$, where $n \in \mathbb{Z}$.

At each such $x = \frac{n}{2}$, the function jumps by 1 and hence is discontinuous.

Therefore, the set of all points of discontinuity is:

$\left\{\,\frac{n}{2} : n \in \mathbb{Z}\,\right\}$

That is, discontinuities occur at $\dots, -1, -\frac{1}{2}, 0, \frac{1}{2}, 1, \frac{3}{2}, 2, \dots$

Final Answer: Set of discontinuities $= \frac{1}{2}\mathbb{Z}$.