Assertion Let = \) +\sin x & 0 \leq x \leq \pi / 2 \\ 3 & x \geq \pi / 2\end{array}\right. \) is continuous on Reason The greatest integer function is discontinuous at every integer.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Maths · Limits, continuity and differentiability

Assertion Let = \) +\sin x & 0 \leq x \leq \pi / 2 \\ 3 & x \geq \pi / 2\end{arr

Assertion Let \( \boldsymbol{f}(\boldsymbol{x})= \) \( \left\{\begin{array}{ll}1+x & x<0 \\ 1+[x]+\sin x & 0 \leq x \leq \pi / 2 \\ 3 & x \geq \pi / 2\end{array}\right. \) is continuous on \( \mathrm{R}-\{1\} \) Reason The greatest integer function is discontinuous at every integer.

A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect

Step-by-step solution

The function f(x) is continuous at all points except x=1 due to the jump in the greatest integer function. At x=1, the left-hand limit is 1+sin1 and right-hand limit is 2+sin1, so not equal, hence discontinuity. The reason correctly states that the greatest integer function is discontinuous at every integer, which explains the discontinuity at x=1. Thus both are correct and reason is the correct explanation.

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Step-by-step solution

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