Consider f(x) = [x] + sqrt{{x}},where [.] den | vedclass

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(D) Let $f(x) = [x] + \sqrt{\{x\}}$.For any non-integer $x$,$[x]$ and $\{x\}$ are continuous,so $f(x)$ is continuous.Now,check the continuity at an in

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$f(x)$ is continuous $\forall x \in R$.

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(D) Let $f(x) = [x] + \sqrt{\{x\}}$.For any non-integer $x$,$[x]$ and $\{x\}$ are continuous,so $f(x)$ is continuous.Now,check the continuity at an integer $n \in I$.Left-hand limit: $\lim_{x 	o n^-} f(x) = \lim_{x 	o n^-} ([x] + \sqrt{\{x\}}) = (n-1) + \sqrt{1} = n-1+1 = n$.Right-hand limit: $\lim_{x 	o n^+} f(x) = \lim_{x 	o n^+} ([x] + \sqrt{\{x\}}) = n + \sqrt{0} = n$.Value of the function: $f(n) = [n] + \sqrt{\{n\}} = n + 0 = n$.Since $\lim_{x 	o n^-} f(x) = \lim_{x 	o n^+} f(x) = f(n) = n$,the function is continuous at all integers.Therefore,$f(x)$ is continuous $\forall x \in R$.

Consider $f(x) = [x] + \sqrt{\{x\}}$,where $[.]$ denotes the greatest integer function and $\{.\}$ denotes the fractional part function. Identify the correct statement.

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