# Only Transcendentals Known

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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