Class 12 Continuity And Differentiability · Long Answer3 Marksdifferentiability
Class 12 Maths · Continuity And Differentiability · Long Answer #63
Show f(x)=|x−3| is continuous but not differentiable at 3.
✅ Step-by-Step Solution
Final Answer: Continuous at 3; not differentiable there.
- The value f(3)=0. For h>0, f(3−h)=h and f(3+h)=h.
- Both values tend to 0, equal to f(3), proving continuity.
- The left quotient is h/(−h)=−1, while the right quotient is h/h=1. Their finite limits disagree.
- Thus f is not differentiable at 3 despite being continuous there.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] The value f(3)=0. For h>0, f(3−h)=h and f(3+h)=h.
- [1 mark] Both values tend to 0, equal to f(3), proving continuity.
- [1 mark] The left quotient is h/(−h)=−1, while the right quotient is h/h=1. Their finite limits disagree.
- [0.5 marks] Thus f is not differentiable at 3 despite being continuous there.
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