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  • Let be a function defined on <img alt="\mathrm{R \: by \: f(x)=[x]+\sqrt{x-[x]}}" src="https://entran

Let \mathrm{f} be a function defined on \mathrm{R \: by \: f(x)=[x]+\sqrt{x-[x]}}, then
 

Option: 1

\mathrm{f} is not continuous at every \mathrm{x \in I}

 


Option: 2

\mathrm{f} is not continuous at every \mathrm{x \in R \sim I}
 


Option: 3

\mathrm{f} is a continuous function
 


Option: 4

\text{none of these}


Answers (1)

best_answer

\mathrm{f(x)=n+\sqrt{x-n}, n \leq x<n+1}

\mathrm{If \: x_0=K \in I, then \lim _{x \rightarrow K+} f(x)=\lim _{x \rightarrow K} K+\sqrt{x-K}=K}

and \mathrm{ \lim _{x \rightarrow K^{-}} f(x)=\lim _{x \rightarrow K^{-}}(K-1)+\sqrt{x-(K-1)} }

\mathrm{ = \lim _{x \rightarrow K}(K-1)+\sqrt{x-(K+1)}=K-1+1=K }

Hence \mathrm{ f } is continuous at every \mathrm{ x_0=K \in I. \, If \: x_0 \in R \sim I, } then \mathrm{ [x] } is continuous at \mathrm{ x_0} which is turn gives that \mathrm{f} is continuous at every \mathrm{x_0 \in R \sim I}

Hence option 3 is correct.

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chirag

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