ENADE Matemática Practice Questions - ENADE Matemática - Exame Nacional De Desempenho Dos Estudantes Exam

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Let $f: [0,1] \to \mathbb{R}$ be Thomae's function (also known as the modified Dirichlet function), defined by $f(x) = 1/q$ if $x = p/q \in \mathbb{Q}$ is in lowest terms ($p \ge 0, q > 0$), $f(0) = 1$, and $f(x) = 0$ if $x \in [0,1] \setminus \mathbb{Q}$. Which of the following statements correctly characterizes the Riemann integrability and integral of $f$ on $[0,1]$?

$$f$ is not Riemann integrable on $[0,1]$ because it is discontinuous on all rational numbers in $[0,1]$. $$f$ is Riemann integrable on $[0,1]$ and its integral equals $1$. $$f$ is continuous everywhere on $[0,1]$, hence Riemann integrable with integral $1/2$. $$f$ is Riemann integrable on $[0,1]$ and $\int_0^1 f(x) dx = 0$ because the set of its discontinuities $\mathbb{Q} \cap [0,1]$ has Lebesgue measure zero.

Correct Answer:

$$f$ is Riemann integrable on $[0,1]$ and $\int_0^1 f(x) dx = 0$ because the set of its discontinuities $\mathbb{Q} \cap [0,1]$ has Lebesgue measure zero.

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