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GATE 2023 ST – Question 45

Calculus · Sequences and Series · 2 marks · Multiple choice

f:[0,1]→ℝ is continuous. Which statement is NOT true?

  1. For every sequence x_n in [0,1],Σ f(x_n)/n² converges absolutely.
  2. If |f(x)|=1 everywhere,then |∫₀¹f(x)dx|=1.
  3. If f(x_n) converges,then x_n converges.
  4. If f is also increasing,its image is [f(0),f(1)].

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Show answer and explanation

Correct answer: (C) If f(x_n) converges,then x_n converges.

Explanation

Continuity on a compact interval makes f bounded,proving A by comparison with Σ1/n². The intermediate value theorem makes a continuous ±1-valued function constant,proving B. It also proves D. For C,take f≡0 and x_n alternating 0,1:images converge but inputs do not. **C**.

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