Question

Let $f\left(x\right)=\left[x\right]+[-x]$. Then for any integer n and $k\in R-I$

• A. $\underset{x\to n}{lim}f\left(x\right)$ exists
• B. $\underset{x\to k}{lim}f\left(x\right)$ exists
• C. f is continuous at $x=n$
• D. f is continuous at $x=k$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $\underset{x\to n}{lim}f\left(x\right)$ exists
B $\underset{x\to k}{lim}f\left(x\right)$ exists
D f is continuous at $x=k$

$f\left(n\right)=\left[n\right]+[-n]=n-n=0$
$\underset{x\to n^{-}}{lim}f\left(x\right)=\underset{h\to 0^{+}}{lim}[n-h]=[-n+h]=n-1+(-n)=-1$
$\underset{x\to n^{+}}{lim}f\left(x\right)=\underset{h\to 0^{+}}{lim}[n+h]=[-n-h]=n+(-n-1)=-1$
If $k\notin I$, the function [x] and $[-x]$ are continuous at k.
Hence $\underset{x\to k}{lim}$ f(x) exists.