In [1,3],the function [x² + 1],[x] denoting greatest integer function ,is discontinuous at..........points.

The greatest integer function is discontinuous at all the integral points.

Thus, $\left[x^2+1\right]$ will be discontinuous if $x^2+1=n, n\in {\rm Z}$

$$\begin{gathered} \Rightarrow x^2+1=n \\ \Rightarrow x=\pm \sqrt{n-1} \end{gathered}$$

Now check the values in the interval $\left[1, 3\right]$, we have
$x=1,\sqrt{2},\sqrt{3},4,\sqrt{5},\sqrt{6},\sqrt{7},\sqrt{8},3$

Thus, the given function is discontinuous at $x=1,\sqrt{2},\sqrt{3},4,\sqrt{5},\sqrt{6},\sqrt{7},\sqrt{8},3$ in $\left[1, 3\right]$