Question

$f\left(x\right)=\frac{1}{q}$ if $x=\frac{p}{q}$ where p and q are integer and $q\ne 0$, G.C.D of (p, q) = 1 and f(x) = 0 if x is irrational then set of continuous points of f(x) is

• A. All real numbers
• B. All rational numbers
• C. All irrational number
• D. All integers

Answer 1

The correct option is C All irrational number

Let $x=\frac{p}{q}$
$f\left(x\right)=\frac{1}{q}$
When $x\to \frac{p}{q}f\left(x\right)=0$ for every irrational number $ϵnbd\left(\frac{p}{q}\right)$
$=\frac{1}{n}ifn=\frac{m}{n}ϵnbd\left(\frac{p}{q}\right)\frac{1}{n}\to 0asn\to \infty sinceThere\infty -numberofrationalϵnbd\left(\frac{p}{q}\right)\therefore \underset{x\to \frac{p}{q}}{lim}f\left(x\right)=0butf\left(\frac{p}{q}\right)=\frac{1}{q}\ne 0DiscontinuousateveryrationalIfx=\alpha isirrational\Rightarrow f\left(\alpha \right)=0Now\underset{x\to \infty }{lim}f\left(x\right)isalso0\therefore continuousforeveryirrational\alpha$