Continuity And Differentiability

Step 1: For continuity at , Step 2: Left-hand limit (LHL): Right-hand limit (RHL): Step 3: So, Therefore, . %Quciktip

To determine the correctness of the Assertion (A) and Reason (R), we analyze the greatest integer function , where denotes the greatest integer less than or equal to . - Assertion (A): , , the greatest integer function is not differentiable at .
The greatest integer function has a step-like graph, with jumps at every integer. Differentiability requires the function to have a well-defined derivative, which involves the existence of left-hand and right-hand derivatives being equal. At :
- Left-hand derivative at : As approaches 2 from the left (e.g., ), , so the difference quotient approaches 0.
- Right-hand derivative at : As approaches 2 from the right (e.g., ), , so the difference quotient approaches infinity.
Since the left-hand and right-hand derivatives are not equal, is not differentiable at . Assertion (A) is true.
- Reason (R): The greatest integer function is not continuous at any integral value.
The greatest integer function has discontinuities at integer values because it jumps from one integer to the next. For example, at :
- As approaches 2 from the left, .
- As approaches 2 from the right, .
- The value at is , so the limit does not exist, and the function is not continuous at . This holds for all integers. Thus, Reason (R) is true.
- Explanation Link: Differentiability requires continuity. Since the greatest integer function is not continuous at (and at any integer), it cannot be differentiable at . Reason (R) explains why Assertion (A) is true, as the discontinuity at integral values prevents differentiability. - Conclusion: Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). The correct option is (A).