Question

The function f(x) = [x] + [-x] is continuous on (where [.] denotes the greatest integer function)

• A. $\mathbb{R}$
• B. $\mathbb{Z}$
• C. $\mathbb{R} \setminus \mathbb{Z}$
• D. $(0,1)$

Answer 1

Answer: (C), (D)

$\mathbb{R} \setminus \mathbb{Z}$

Answer 2

For any integer $x=n$, $f(n) = [n] + [-n] = n + (-n) = 0$. For any non-integer $x$, let $x = n + \alpha$ where $n \in \mathbb{Z}$ and $0 < \alpha < 1$. Then $f(x) = [n+\alpha] + [-n-\alpha] = n + (-n-1) = -1$. Thus, $f(x) = 0$ if $x \in \mathbb{Z}$ and $f(x) = -1$ if $x \notin \mathbb{Z}$. At integer points $x=n$, the left-hand limit is -1 and the right-hand limit is -1, but $f(n)=0$, so it's discontinuous. At non-integer points, the function is constantly -1, hence continuous.