Question

$f\left(x\right)=\left[x\right]$ is discontinuous at $x=1$ because

• A. Value of function is not defined
• B. Value of $f\left(x\right)$ is finite but not equal to the limit
• C. Limit doesn't exist
• D. Statement wrong, function is continuous

Answer 1

The correct option is C

Limit doesn't exist

Lets try to draw the greatest integer function.

Greatest integer function is also called step function where its breaks at each integer. This type of sudden breaks comes under non removable discontinuity of the second type.

Here,

$\underset{x\to 1^{-}}{lim}f\left(x\right)=\underset{x\to 1^{-}}{lim}\left[x\right]=0$

and $\underset{x\to 1^{+}}{lim}f\left(x\right)=\underset{x\to 1^{+}}{lim}\left[x\right]=1$

Since L.H.L $\ne$ R.H.L we can say the limit doesn't exist.