Question

# Prove that the greatest integer function is defined by,$f(x) = [x],0 < x < 3$ is not differentiable at $x{\text{ }} = {\text{ }}1$ and $x{\text{ }} = {\text{ 2}}$.

Hint: In this question first we will check the continuity of the given function at a given point, if it is discontinuous at that given point, it will also be non-differentiable at that point.

As we are given with a function,
$\Rightarrow f(x) = [x],0 < x < 3$
$\Rightarrow$As, we know that, if ${\text{ }}f(x)$ is not continuous at a point,
Then it will not be differentiable at that point too.
$\Rightarrow$So, let us check for continuity of $f(x){\text{ }} = {\text{ }}\left[ x \right]$ at $x{\text{ }} = {\text{ }}1$ and $x{\text{ }} = {\text{ 2}}$.
Checking continuity at $x{\text{ }} = {\text{ }}1$
$\Rightarrow \mathop {\lim }\limits_{x \to {1^ - }} f(x) = \mathop {\lim }\limits_{x \to {1^ - }} [x] = 0$
$\Rightarrow$And, $\mathop {\lim }\limits_{x \to {1^ + }} f(x) = \mathop {\lim }\limits_{x \to {1^ + }} [x] = 1$
As, we have seen above that $\mathop {\lim }\limits_{x \to {1^ - }} f(x) \ne \mathop {\lim }\limits_{x \to {1^ + }} f(x)$
$\Rightarrow$Therefore, ${\text{ }}f(x)$ is neither continuous nor differentiable at $x{\text{ }} = {\text{ }}1$.
Now, checking continuity at ${\text{x}} = 2$.
$\Rightarrow \mathop {\lim }\limits_{x \to {2^ - }} f(x) = \mathop {\lim }\limits_{x \to {2^ - }} [x] = 1$
$\Rightarrow$And, $\mathop {\lim }\limits_{x \to {2^ + }} f(x) = \mathop {\lim }\limits_{x \to {2^ + }} [x] = 2$
As, we have seen above that, $\mathop {\lim }\limits_{x \to {2^ - }} f(x) \ne \mathop {\lim }\limits_{x \to {2^ + }} f(x)$.
$\Rightarrow$Therefore, ${\text{ }}f(x)$ is neither continuous nor differentiable at ${\text{x}} = 2$.
$\Rightarrow$Hence, ${\text{ }}f(x)$ is neither differentiable at $x{\text{ }} = {\text{ }}1$
nor differentiable at ${\text{x}} = 2$.

Note: Whenever we come up with this type of problem we are asked to check whether the function, ${\text{ }}f(x)$ is differentiable or not. Then first we should check the continuity of the given function, if it is continuous then we have to check whether the function is differentiable or not.