Question

Examine the continuity of the function $f\left(x\right)=\left[x\right]$, where $\left[x\right]$ denotes the greatest integer less then or equal to $x$.

Answer 1

We have the given function

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

$\underset{x\to n^{-}}{lim}\left[x\right]=n-1\underset{x\to n^{+}}{lim}\left[x\right]=x$

As limit does not exist .

Hence, $\left[x\right]$ is discontinuous at every integer.

So, $\left[x\right]$ is continuous at any non integer value of $x$.