Question

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

Answer 1

At $x=1$;

$f\left(x\right)$ is differentiate at $x=1$ if

$LHD=RHD$

$\therefore LHD$

${lim}_{h\to 0}\frac{f\left(1\right)-f(1-h)}{h}={lim}_{h\to 0}\frac{f\left(1\right)-f(1-h)}{h}={lim}_{h\to 0}\frac{\left[1\right]-[1-h]}{h}={lim}_{h\to 0}\frac{1-0}{h}={lim}_{h\to 0}\frac{1}{h}$

$=\frac{1}{0}$ not defined

$RHD{lim}_{h\to 0}\frac{f(1+h)-f\left(1\right)}{h}={lim}_{h\to 0}\frac{1-1}{h}=0$

However $LHD\ne RHD$

Hence the function is not differentiable at $x=0$.