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Provide solution for rd sharma maths class 12 chapter 9 Differentiability exercise Multiple choice question, question 14

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HINTS: understand the continuity and differentiability of [x]

GIVEN: f(x)=x-[x]



f(x)=\left\{\begin{array}{cc} x-[x] & x \neq n \text { when } n \in Z \\ 0 & x \in n \end{array}\right.


LHL at x=n where n \in Z \\

 \begin{aligned} \lim _{x \rightarrow n^{-}} f(x) &=\lim _{x \rightarrow n} n-(n-1) \quad\{[n]=n-1\\ &=\lim _{x \rightarrow n} 1 \\ &=1 \end{aligned}

RHL at z=n

\begin{aligned} \lim _{x \rightarrow n^{+}} f(x)=& \lim _{x \rightarrow n} n-n=0 \quad\{[n]=n\\ & \lim _{x \rightarrow n^{-}} f(x) \text { and } \lim _{x \rightarrow n^{+}} 1 f(x) \end{aligned}

This f is not continues at integer points. Hence f is continuous at non interger points only

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