Only Transcendentals Known

Calculus Level 4

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

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

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

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