ThmDex – An index of mathematical definitions, results, and conjectures.
Riemann integral of a constant function on the closed unit interval
Formulation 0
Let $f : [0, 1] \to \mathbb{R}$ be a D4364: Real function such that
(i) \begin{equation} \exists \, \lambda \in \mathbb{R} : \forall \, x \in [0, 1] : f(x) = \lambda \end{equation}
Then
(1) $f$ is a D1760: Riemann integrable real function
(2) \begin{equation} \int^1_0 f(x) \, d x = \lambda \end{equation}
Proofs
Proof 0
Let $f : [0, 1] \to \mathbb{R}$ be a D4364: Real function such that
(i) \begin{equation} \exists \, \lambda \in \mathbb{R} : \forall \, x \in [0, 1] : f(x) = \lambda \end{equation}
This result is a particular case of R2113: Riemann integral of a constant function. $\square$