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need solution for RD Sharma maths class 12 chapter 8 Continuity exercise Fill in the blanks question 11

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Answer: All integral points

Hint: \left [ x \right ] is called greatest integer function

Given:  f(x)=x-\left [ x \right ] is discontinuous


            Let c be an integer and g(x)=x-[x]

            g(x) is continuous at x=c ,if LHL=RHL=g(c)

            \begin{aligned} &\Rightarrow \lim _{x \rightarrow c^{-}} g(x)=\lim _{x \rightarrow c^{+}} g(x)=g(c)\\ &\text { LHL, }\\ &\lim _{x \rightarrow c^{-}} g(x)=\lim _{h \rightarrow 0} g(c-h) \end{aligned}

            \begin{aligned} &=\lim _{h \rightarrow 0} c-h-[c-1] \\ &=\lim _{h \rightarrow 0}(c-h)-[c-1] \\ &=\lim _{h \rightarrow 0}-h+1 \\ &=-0+1 \\ &=1 \end{aligned}

            \begin{aligned} &\mathrm{RHL}, \\ &\lim _{x \rightarrow c^{+}} g(x)=\lim _{h \rightarrow 0} g(c+h) \\ &=\lim _{h \rightarrow 0} c+h-[c+2h] \\ &=\lim _{h \rightarrow 0} c-h-c \\ &=\lim _{h \rightarrow 0}-h \\ &=0 \end{aligned}

            LHL\neq RHL

            Therefore, g(x) is not continuous.



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