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Part CCSIR NET December 2025all-derivatives-vanishing-at-a-point-does-not-make-a-function-analytic-there-e-to-the-minus-1-over-x-squared-is-the-standard-counterexample

All derivatives vanishing at a point does not make a function analytic there e to the minus 1 over x squared is the standard counterexample

Consider the following statements: (P) The initial value problem y′+y=e(−x2),y(0)=0y'+y=e^(-x^{2}), y(0)=0 has a Taylor series solution about the point x=0. (Q) The initial value problem y′+y=r(x), y(0)=0, where r(x)=e(−1/x2)r(x)=e^(-1/x^{2}) for x≠0 and r(0)=0, has a Taylor series solution about the point x=0. Then which of the following statements are true?

  1. A.(P) is true.✓ Correct
  2. B.(P) is FALSE.
  3. C.(Q) is true.
  4. D.(Q) is FALSE.✓ 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: Zeros of a non-constant holomorphic function cannot accumulate

More on this topic

The chapter behind this: Analyticity, the identity theorem and zeros — free to read

From Analytic Functions › Power series and analyticity

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