Question

For the options given below, the function f(x) = [x] is discontinuous at x =

• A. $1$
• B. $-0.5$
• C. $0.5$
• D. $1.5$

Answer 1

The correct option is A $1$

Lets try to draw the greatest integer function.

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

Here

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

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

Since LHL $\ne$ RHL we can say the limit doesn't exist.
So the function is always discontinuous at all integer points.