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Continuity: The function $f(x) = |x|$ is continuous everywhere. This can be visualized by its graph, which has no breaks.
Differentiability: The function $f(x) = |x|$ is not differentiable at $x=0$ because the left-hand derivative and the right-hand derivative at $x=0$ are not equal. For $x < 0$, $f(x) = -x$, so $f'(x) = -1$. For $x > 0$, $f(x) = x$, so $f'(x) = 1$. At $x=0$, the derivative is undefined.
Conclusion: Therefore, $f(x) = |x|$ is continuous everywhere, but differentiable everywhere except at $x=0$.
Correct Answer: continuous everywhere, but differentiable everywhere except at $x=0$<\/strong>
AI Suggestion: Option C
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Pedagogical Audit
Bloom's Analysis:
This is an UNDERSTAND question because it requires students to recall the properties of the modulus function, specifically its continuity and differentiability.
Knowledge Dimension:CONCEPTUAL
Justification:The question tests the understanding of the concept of continuity and differentiability of a modulus function. It requires the student to know the general properties of such functions rather than specific facts or procedures.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly assesses the understanding of continuity and differentiability, which are core concepts covered in the textbook.