Let $f(x)=(-1)^{\left[x^3\right]}$, where [.] denotes the greatest integer function. Then,

f(x) is discontinuous at $x=n^{1 / 3}, n \in Z$

We observe that $x^3$ attains integral values at $x=n^{1 / 3}$, where $n \in Z$. Therefore,

$f(x)=(-1)^{\left[x^3\right]}$

$\Rightarrow f(x)=(-1)^n$ for $x \in\left[n^{1 / 3},(n+1)^{1 / 3}\right)$, where $n \in Z$

$\Rightarrow f(x)=\left\{\begin{array}{l}1, \text { if } x \in\left[(2 n)^{1 / 3},(2 n+1)^{1 / 3}\right) \\ -1, \text { if } x \in\left[(2 n+1)^{1 / 3},(2 n+3)^{1 / 3}\right) \end{array}\right.$, where $n \in Z$

Clearly, f(x) is discontinuous at $x=n^{1 / 3}, n \in Z$.