Question

The function $y=1+\log \left[x\right]$ is ($\left[x\right]$ is the greatest integer function)

• A. continuous nowhere
• B. continuous everywhere
• C. continuous at infinitely many points
• D. discontinuous at infinitely many points

Answer 1

Answer: (C), (D)

discontinuous at infinitely many points
continuous at infinitely many points

Given: $y=1+\log x$ $\left[x\right]=G.I.F$

To find whether y is continous or not

Sol: Let $f\left(x\right)=\left[x\right]$ and

$g\left(x\right)=1+\log \left(x\right)$

We know that $g\left(f\left(x\right)\right)$ is discontinous if

$f\left(x\right)$ is discontinous

$f\left(x\right)=\left[x\right]$ is discontinous at many points i.e all integral values of x

Hence $g\left(f\left(x\right)\right)=1+\log \left(x\right)$ is also discontinous at infinitely many points