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NCERT Class 12 · Ch 5 Miscellaneous Exercise · Q20

NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Miscellaneous Exercise Question 20

Does an everywhere-continuous function exist which is not differentiable at exactly two points? Justify.
Exactly two corners
Exactly two corners

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: Yes: f(x)=|x|+|x−1|, with corners exactly at 0 and 1.
  1. It is a sum of continuous absolute-value functions, hence continuous on R.
  2. Its formulas are 1−2x for x<0, 1 for 0≤x≤1, and 2x−1 for x>1. Slopes are −2,0,2.
  3. At 0 and 1 the adjacent slopes differ; at every other point there is a linear neighbourhood and a derivative exists.

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