1. ## discontinuity of function

points of discontinuity for the function

f(x) = x + [x] (bracket function)

2. Analyze the cases $\displaystyle a\in \mathbb{Z}$ and $\displaystyle a\not\in \mathbb{Z}$ .

3. As per your guidance f(x) is discont as x belongs to Z but how we analyze for x doesnot belong to Z

4. Suppose $\displaystyle a\in(n,n+1)$ with $\displaystyle n\in\mathbb{Z}$ then, in a neighborhood $\displaystyle V(a)$ of $\displaystyle a$ we have $\displaystyle f(x)=x+\lfloor x \rfloor=x+n$ .