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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability3 Marks
Question
Prove that the greatest integer function defined by $f\left( x \right) = \left[ x \right],0 < x < 3$ is not differentiable at x = 1 and x = 2.
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Answer

Given: $f\left( x \right) = \left[ x \right],0 < x < 3$
$Rf'\left( 1 \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {1 + h} \right) - f\left( 1 \right)}}{h}$
= $\mathop {\lim }\limits_{h \to 0} \frac{{\left[ {1 + h} \right] - 1}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{1 - 1}}{h}$
= $\mathop {\lim }\limits_{h \to 0} \frac{0}{h}$ = 0
And $Lf'\left( 1 \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {1 - h} \right) - f\left( 1 \right)}}{{ - h}}$

$= \mathop {\lim }\limits_{h \to 0} \frac{{\left[ {1 - h} \right] - 1}}{{ - h}}$

$ = \mathop {\lim }\limits_{h \to 0} \frac{{0 - 1}}{{ - h}} = \infty $
Since $Rf'\left( 1 \right) \ne Lf'\left( 1 \right)$
Therefore, $f\left( x \right) = \left[ x \right]$ is not differentiable at x =1.

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