Only Transcendentals Known

Calculus Level 4

Consider all functions \( f: \mathbb{R} \to \mathbb{R} \) satisfying that

  • \(f(x) = 0\) for all transcendental numbers \(x\),
  • the Lebesgue-integral \( \displaystyle \int_{-\infty}^{\infty} f(t) \, dt \) exists.

Let \( S = \left\{ \displaystyle \int_{-\infty}^{\infty} f(t) \, dt \right\} \). Which of the following describes \( S \)?

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