Q1: What is the correct definition for a bounded function $f$ being Riemann-integrable?

A1: $$ \sup \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi \leq f \right} = \inf \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi \geq f \right} $$

A2: $$ \sup \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi \geq f \right} = \inf \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi \leq f \right} $$

A3: $$ \sup \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi \geq f \right} < \inf \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi < f \right} $$

A4: $$ \sup \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi \geq f \right} > \inf \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi > f \right} $$

Q2: Let $\psi : [0,2] \rightarrow \mathbb{R}$ be a step function. Is $\psi$ Riemann-integrable?

A1: Yes!

A2: No!