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

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

Solution:

The provided function is $f(x)=[x], 0<x<3$

R.H.L. :

$f^{\prime}(1)=\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}$

$\lim _{h \rightarrow 0} \frac{|1+h|-1}{h}$

$\lim _{h \rightarrow 0} \frac{1-1}{h}$

$=\lim _{\lambda \rightarrow 0} \frac{0}{h}=0$

and L.H.L. :

$f^{\prime}(1)=\lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{-h}$

$\lim _{=h \rightarrow 0} \frac{|1-h|-1}{-h}$

$\lim _{=\lambda \rightarrow 0} \frac{0-1}{-h}=\infty$

R.H.L. $\neq$ L.H.L.

As a result, $f(x)=[x]$ is not differentiable at $x=1$ .

Similarly, $f(x)=[x]$ is not differentiable at $x=2$ .

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