Q1: Let $f: [0,1] \rightarrow \mathbb{R}$ be Riemann-integrable. Which conclusion is correct?

A1: $f$ is continuous.

A2: $f$ is monotonic.

A3: $f$ is bounded.

Q2: $f: [0,1] \rightarrow [0,1]$ be given by $$ f(x) = \begin{cases} x^2, & x > 0 \ 1, & x = 0 \end{cases}$$ Is this function Riemann-integrable?

A1: Yes!

A2: No!

A3: Only as an improper Riemann integral.

Q3: $f: [0,1] \rightarrow [0,1]$ be given by $$ f(x) = \begin{cases} \frac{1}{x}, & x > 0 \ 1, & x = 0 \end{cases}$$ Does the integral $$\int_0^1 f(x) , dx$$ exist?

A1: No, the improper Riemann integral does not converge.

A2: Yes, it’s a well-defined Riemann integral.

A3: Yes, but only as an improper Riemann integral.