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Let \mathrm{f(x)=\left[x^2\right]-[x]^2}, where [\cdot]  denotes the greatest integer function. Then
 

Option: 1

\mathrm{f(x)} is discontinuous for all integral values of \mathrm{ x}
 


Option: 2

\mathrm{f(x) }is discontinuous only at \mathrm{x=0,1}
 


Option: 3

\mathrm{f(x) }is continuous only at \mathrm{x=1}
 


Option: 4

none of these


Answers (1)

best_answer

\begin{aligned} & \mathrm{f(1+0)=\lim _{h \rightarrow 0}\left\{\left[(1+h)^2\right]-[1+h]^2\right\}=\lim _{h \rightarrow 0}\{1-1\}=0 . }\\ &\mathrm{ f(1-0)=\lim _{h \rightarrow 0}\left\{\left[(1-h)^2\right]-[1-h]^2\right\}=\lim _{h \rightarrow 0}\{0-0\}=0 }. \\ &\text{Also f(1)=0. So, f(x) is continuous at x=1.}\end{aligned}
 

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Kuldeep Maurya

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