# Only Transcedentals Known

Calculus Level 5

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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