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GATE 2024 AE – Question 45

Engineering Mathematics · Calculus: Limits, continuity, differentiability, chain rule, maxima and minima, integration · 2 marks · Multiple select

Consider the function

$f(x) = \begin{cases} x^2 & \text{for } x < 0 \\ x & \text{for } x \ge 0 \end{cases}$

where x is real. Which of the following statements is/are correct?

  1. The function is continuous for all x
  2. The derivative of the function is discontinuous at x = 0
  3. The derivative of the function is continuous at x = 1
  4. The function is discontinuous at x = 0

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Show answer and explanation

Correct answer: (A) The function is continuous for all x; (B) The derivative of the function is discontinuous at x = 0; (C) The derivative of the function is continuous at x = 1

Explanation

Both pieces tend to 0 at $x = 0$, so $f$ is continuous everywhere. The derivative is $2x \to 0$ from the left and 1 from the right at $x = 0$, so it is discontinuous there. Near $x = 1$ the derivative is the constant 1, which is continuous.