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Class 12 Continuity And Differentiability · Long Answer3 Markspiecewise_continuity

Class 12 Maths · Continuity And Differentiability · Long Answer #126

For f(x)=x−|x−x²| on [−1,1], find every point of discontinuity, using relative continuity at endpoints.

✅ Step-by-Step Solution

Final Answer: No discontinuities on [−1,1].
  1. On [−1,0], x−x²≤0, so f(x)=x+(x−x²)=2x−x².
  2. On [0,1], x−x²≥0, so f(x)=x−(x−x²)=x².
  3. Both polynomials have value 0 at the junction 0; each is continuous on its closed piece and the endpoint one-sided limits match f(−1),f(1).
  4. Thus f is continuous everywhere on [−1,1]; there are no discontinuity points.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] On [−1,0], x−x²≤0, so f(x)=x+(x−x²)=2x−x².
  • [1 mark] On [0,1], x−x²≥0, so f(x)=x−(x−x²)=x².
  • [1 mark] Both polynomials have value 0 at the junction 0; each is continuous on its closed piece and the endpoint one-sided limits match f(−1),f(1).
  • [0.5 marks] Thus f is continuous everywhere on [−1,1]; there are no discontinuity points.

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