Question

Assertion (A): \( f(x) = [x] \), \( x \in \mathbb{R} \), the greatest integer function is not differentiable at \( x = 2 \). Reason (R): The greatest integer function is not continuous at any integral value.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

A function is differentiable at a point only if it is continuous there. The greatest integer function's jumps at integers cause both discontinuity and non-differentiability.

Answer 1

The Correct Option is A

To determine the correctness of the Assertion (A) and Reason (R), we analyze the greatest integer function \( f(x) = [x] \), where \( [x] \) denotes the greatest integer less than or equal to \( x \). - Assertion (A): \( f(x) = [x] \), \( x \in \mathbb{R} \), the greatest integer function is not differentiable at \( x = 2 \).
The greatest integer function \( f(x) = [x] \) 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 \( x = 2 \):
- Left-hand derivative at \( x = 2 \): As \( x \) approaches 2 from the left (e.g., \( x = 1.9, 1.99 \)), \( f(x) = 1 \), so the difference quotient approaches 0.
- Right-hand derivative at \( x = 2 \): As \( x \) approaches 2 from the right (e.g., \( x = 2.1, 2.01 \)), \( f(x) = 2 \), so the difference quotient approaches infinity.
Since the left-hand and right-hand derivatives are not equal, \( f(x) = [x] \) is not differentiable at \( x = 2 \). Assertion (A) is true.
- Reason (R): The greatest integer function is not continuous at any integral value.
The greatest integer function \( f(x) = [x] \) has discontinuities at integer values because it jumps from one integer to the next. For example, at \( x = 2 \):
- As \( x \) approaches 2 from the left, \( f(x) \to 1 \).
- As \( x \) approaches 2 from the right, \( f(x) \to 2 \).
- The value at \( x = 2 \) is \( f(2) = 2 \), so the limit does not exist, and the function is not continuous at \( x = 2 \). This holds for all integers. Thus, Reason (R) is true.
- Explanation Link: Differentiability requires continuity. Since the greatest integer function is not continuous at \( x = 2 \) (and at any integer), it cannot be differentiable at \( x = 2 \). 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). \[ \boxed{\text{(A)}} \]