Question

The function $f\left(x\right)=\left(x\right),$ where $\left(x\right)$ denotes the smallest integer $\ge x,$ is

• A. Continuous at integral points
• B. Continuous at non integral points
• C. Discontinuous at integral points
• D. Discontinuous at non-integral points

Answer 1

Answer: (B), (C)

Continuous at non integral points
Discontinuous at integral points

Let $x=n,n\in Z$

Then, $L.H.L.=\underset{x\to n;x<n}{lim}\left(x\right)=n;R.H.L.=\underset{x\to n;x>n}{lim}\left(x\right)=n+1$

Since, $L.H.L.\ne R.H.L.,$ therefore $f\left(x\right)$ is discontinuous at all integers $n.$

Now, let $x=p,n<p\le n+1,$ where $n$ is an integer.

Then, $L.H.L.=\underset{x\to p;x<p}{lim}\left(x\right)=n+1$

$R.H.L.=\underset{x\to p;x>p}{lim}\left(x\right)=n+1$

$f\left(p\right)=\left(p\right)=n+1$

Since, $L.H.L.=R.H.L.=f\left(p\right),$

therefore, $f\left(x\right)$ is continuous at all non-integral points $p.$