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Consider the following statements:
1- The function $\mathrm{f}(\mathrm{x})=$ greatest integer $\leq \mathrm{x}, \mathrm{x} \in \mathrm{R}$ is a continuous function.
2- All trigonometric functions are continuous on $\mathrm{R}$. Which of the statements given above is/are correct?
Options:
1- The function $\mathrm{f}(\mathrm{x})=$ greatest integer $\leq \mathrm{x}, \mathrm{x} \in \mathrm{R}$ is a continuous function.
2- All trigonometric functions are continuous on $\mathrm{R}$. Which of the statements given above is/are correct?
Solution:
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Verified Answer
The correct answer is:
Neither 1 nor 2
Here, greatest integer function [x] is discontinuous at its integral value of $\mathrm{x}, \cot \mathrm{x}$ and $\operatorname{cosec} \mathrm{x}$ are discontinuous at $0, \pi, 2 \pi$ etc. and $\tan \mathrm{x}$ and sec $\mathrm{x}$ are discontinuous at
$\mathrm{x}=\frac{\pi}{2}, \frac{3 \pi}{2}, \frac{5 \pi}{2}$ etc. Therefore the greatest integer function and all trigonometric functions are not continuous for $\mathrm{x} \in \mathrm{R}$ Therefore, neither (1) nor (2) are true.
$\mathrm{x}=\frac{\pi}{2}, \frac{3 \pi}{2}, \frac{5 \pi}{2}$ etc. Therefore the greatest integer function and all trigonometric functions are not continuous for $\mathrm{x} \in \mathrm{R}$ Therefore, neither (1) nor (2) are true.
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