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 \[x{\text{ }} = {\text{ 2}}\].

Answer 1

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.

Complete step-by-step answer:
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.