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Consider the following statements:
1- The function $f(x)=[x]$, where $[.]$ is the greatest integer function defined on $R$, is continuous at all points except at $x=0 . \quad[2014-I I]$
2- The function $f(x)=\sin |x|$ is continuous for all $x \in R$.
Which of the above statements is/are correct?
Options:
1- The function $f(x)=[x]$, where $[.]$ is the greatest integer function defined on $R$, is continuous at all points except at $x=0 . \quad[2014-I I]$
2- The function $f(x)=\sin |x|$ is continuous for all $x \in R$.
Which of the above statements is/are correct?
Solution:
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Verified Answer
The correct answer is:
2 only
The greatest integer function is continuous at all statement points except integer. Hence, statement 1is incorrect. Statement $2:$ Let $h(a)=\sin x$ and $g(x)=|x|$
$\operatorname{hog}(x)=\sin \mid x$
$\Rightarrow \mathrm{f}(\mathrm{x})=\operatorname{hog}(\mathrm{x})=\sin |\mathrm{x}|$
Therefore, $\mathrm{g}(\mathrm{x})$ is continuous, $\forall \mathrm{x} \in \mathrm{R}$ and $\mathrm{h}(\mathrm{x})$ is continous $\forall \mathrm{x} \in \mathrm{R}$
When both are continuous then $\operatorname{hog}(\mathrm{x})$ is also continuous. Thus, statement 2 is correct.
$\operatorname{hog}(x)=\sin \mid x$
$\Rightarrow \mathrm{f}(\mathrm{x})=\operatorname{hog}(\mathrm{x})=\sin |\mathrm{x}|$
Therefore, $\mathrm{g}(\mathrm{x})$ is continuous, $\forall \mathrm{x} \in \mathrm{R}$ and $\mathrm{h}(\mathrm{x})$ is continous $\forall \mathrm{x} \in \mathrm{R}$
When both are continuous then $\operatorname{hog}(\mathrm{x})$ is also continuous. Thus, statement 2 is correct.
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