Question

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

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

Answer 1

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

$\underset{x\to n}{lim}f\left(x\right)$ exist
$\underset{x\to n}{lim}f\left(k\right)$ exist
f is continuous at k.

We know that:
$[-x]=-\left[x\right]-1,x>0,x\notin I[-x]=-\left[x\right],x\in I$.
So, for every non-integer value of $x$, the function has the value $1$, whereas for every integer value of $x$, the function has value $0$.
$\underset{x\to n}{lim}f\left(x\right)=-1(\because x\notin Iwhenx\to n)$
$\underset{x\to n}{lim}f\left(k\right)=-1(\because k\notin I)$
$f$ is continuous at $k$. It is discontinuous only at integer points.