Let [t] denote the greatest integer
Question:

Let $[t]$ denote the greatest integer $\leq t$. The number of points where the function

$$f(x)=[x]\left|x^{2}-1\right|+\sin \left(\frac{\pi}{[x]+3}\right)-[x+1], x \in(-2,2)$$ is not continuous is

Solution:

$f(x)=[x]\left|x^{2}-1\right|+\sin \frac{\pi}{[x+3]}-[x+1]$

$f(x)=\left\{\begin{array}{cc}3-2 x^{2}, & -2<x<-1 \\ x^{2}, & -1 \leq x<0 \\ \frac{\sqrt{3}}{2}+1 & 0 \leq x<1 \\ x^{2}+1+\frac{1}{\sqrt{2}}, & 1 \leq x<2\end{array}\right.$

discontinuous at $x=0,1$