Question

Consider $f\left(x\right)=x^2;xϵQ$ and $f\left(x\right)=1;x\notin Q$ ; and $g\left(x\right)=\left[x\right]$; where [ ] denotes greatest integer function. Then

• A. $f\left(x\right)$ is continuous at exactly two points
• B. $g\left(x\right)$discontinuous at all $xϵI$
• C. $fog\left(x\right)$ is discontinuous at all $xϵQ$
• D. $gof\left(x\right)$ is continuous everywhere except at $x=$integer

Answer 1

Answer: (A), (B)

The correct options are
A $f\left(x\right)$ is continuous at exactly two points
B $g\left(x\right)$discontinuous at all $xϵI$

$forf\left(x\right)tobecontinuous{x}^2=1\Rightarrow x^2-1=0\Rightarrow x=1,-1$

so f(x) is continuous at 2 points

$g\left(x\right)=\left[x\right]$

g(x) is discontinuous at all integers because

$\underset{x\to k^{-}}{lim}g\left(x\right)=k-1$

$\underset{x\to k^{+}}{lim}g\left(x\right)=k$

Hence answer is A & B