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 $S$ ?

S

is empty

S

is a singleton set

S

is a finite set with at least two elements

S

is a countably infinite set

S

is a uncountablly infinite set