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Part CCSIR NET June 2024partials-exist-without-continuity

Partials exist without continuity

Define f:R2→Rf : \mathbb{R}^{2} \to \mathbb{R} by f(x,y)=yx2+y2/xf(x, y) = y\sqrt{x^{2} + y^{2}}/x if x ≠ 0, and f(x, y) = 0 if x = 0. Which of the following statements are true?

  1. A.∂f/∂x(0,0)\partial{}f/\partial{}x(0, 0) exists✓ Correct
  2. B.∂f/∂y(0,0)\partial{}f/\partial{}y(0, 0) exists✓ Correct
  3. C.f is not continuous at (0, 0)✓ Correct
  4. D.f is not differentiable at (0, 0)✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: If all partial derivatives exist at a point then f is continuous there

More on this topic

The chapter behind this: Differentiability in several variables — free to read

From Functions of Several Variables › Partial derivatives, differentiability, chain rule

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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