Question

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

Answer 1

Hint: To show that the functions are not differentiable at the given points we need to prove that they not continuous at the given points using the condition $\underset{x\to x^{-}}{lim}f(x)=\underset{x\to x^{+}}{lim}f(x)$

Complete step-by-step answer:
We know that a function is differentiable only if it is continuous .
So it is enough if we prove that the function is not continuous at x = 1 and x = 2
For a function to be continuous the left hand derivative must be equal to the right hand derivative
That is , $\underset{x\to x^{-}}{lim}f(x)=\underset{x\to x^{+}}{lim}f(x)$
So let's check the continuity at x = 1
$\Rightarrow \underset{x\to 1^{-}}{lim}f(x)=\underset{x\to 1^{-}}{lim}\left[0\right]=0$
$\Rightarrow \underset{x\to 1^{+}}{lim}f(x)=\underset{x\to 1^{+}}{lim}\left[1\right]=1$
Hence we can see that $\underset{x\to x^{-}}{lim}f(x)\ne \underset{x\to x^{+}}{lim}f(x)$
Hence it is not continuous at x = 1
Hence it is not differentiable at x = 2
So let's check the continuity at x = 2
$\Rightarrow \underset{x\to 2^{-}}{lim}f(x)=\underset{x\to 2^{-}}{lim}\left[1\right]=1$
$\Rightarrow \underset{x\to 2^{+}}{lim}f(x)=\underset{x\to 2^{+}}{lim}\left[2\right]=2$
Hence we can see that $\underset{x\to x^{-}}{lim}f(x)\ne \underset{x\to x^{+}}{lim}f(x)$
Hence it is not continuous at x = 2
Hence it is not differentiable at x = 2
Therefore it is proved that the function is not differentiable at x = 1 and x = 2.

Note: Continuity of a function is the characteristic of a function by virtue of which, the graphical form of that function is a continuous wave. A differentiable function is a function whose derivative exists at each point in its domain.
Every differentiable function is continuous but that doesn’t mean all the continuous functions are differentiable