Class CBSE Class 12 Mathematics Continuity and Differentiability Q #594
KNOWLEDGE BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
If \(f(x)=\{[x],x\in R\}\) is the greatest integer function, then the correct statement is:
(A) f is continuous but not differentiable at \(x=2\).
(B) f is neither continuous nor differentiable at \(x=2\).
(C) f is continuous as well as differentiable at \(x=2\).
(D) f is not continuous but differentiable at \(x=2\).
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Detailed Solution

Step 1: Understanding the Greatest Integer Function

The greatest integer function, denoted by $[x]$, returns the largest integer less than or equal to $x$. For example, $[2.3] = 2$, $[2] = 2$, and $[-1.5] = -2$.

Step 2: Checking Continuity at \(x=2\)

To check continuity at $x=2$, we need to evaluate the left-hand limit (LHL), right-hand limit (RHL), and the function's value at $x=2$. $$LHL = \lim_{x \to 2^-} [x] = 1$$ $$RHL = \lim_{x \to 2^+} [x] = 2$$ $$f(2) = [2] = 2$$ Since $LHL \neq RHL$, the function is not continuous at $x=2$.

Step 3: Checking Differentiability at \(x=2\)

Since the function is not continuous at $x=2$, it cannot be differentiable at $x=2$. Differentiability requires continuity.

Step 4: Conclusion

The function $f(x) = [x]$ is not continuous and therefore not differentiable at $x=2$.

Final Answer: f is neither continuous nor differentiable at \(x=2\).

AI Suggestion: Option B

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of continuity and differentiability to the given greatest integer function.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concepts of continuity and differentiability, and applying them to a specific function (the greatest integer function). It's not just recalling facts, but applying understanding.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of continuity and differentiability of functions, a core topic in calculus.
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