Class CBSE Class 12 Mathematics Continuity and Differentiability Q #596
KNOWLEDGE BASED
UNDERSTAND
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
The function f defined by \(f(x)=\begin{cases}x,&if~x\le1\\ 5,&if~x>1\end{cases}\) is not continuous at:
(A) \(x=0\)
(B) \(x=1\)
(C) \(x=2\)
(D) \(x=5\)
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Correct Answer: B
Explanation
for x=1, LHL=1 and RHL=5

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Detailed Solution

Step 1: Check continuity at x=0

Since $x=0 \le 1$, $f(0) = 0$. The function is defined as $f(x) = x$ in the neighborhood of $x=0$. Therefore, the function is continuous at $x=0$.

Step 2: Check continuity at x=1

We need to check the left-hand limit (LHL) and the right-hand limit (RHL) at $x=1$. LHL: $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x = 1$ RHL: $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} 5 = 5$ Since LHL $\ne$ RHL, the function is not continuous at $x=1$.

Step 3: Check continuity at x=2

Since $x=2 > 1$, $f(2) = 5$. The function is defined as $f(x) = 5$ in the neighborhood of $x=2$. Therefore, the function is continuous at $x=2$.

Step 4: Check continuity at x=5

Since $x=5 > 1$, $f(5) = 5$. The function is defined as $f(x) = 5$ in the neighborhood of $x=5$. Therefore, the function is continuous at $x=5$.

Step 5: Conclusion

The function is not continuous at $x=1$.

Final Answer: x=1

AI Suggestion: Option B

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the definition of continuity and apply it to the given piecewise function.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concept of continuity, limits, and how they apply to piecewise functions.
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, a core concept in the syllabus.
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