$$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\FF}{\mathbb{F}}
\renewcommand{\epsilon}{\varepsilon}
% ALTERNATE VERSIONS
% \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}}
% \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}}
% \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}}
% \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}}
% \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}}
\newcommand{\uppersum}[1]{U_{#1}}
\newcommand{\lowersum}[1]{L_{#1}}
\newcommand{\upperint}[1]{U_{#1}}
\newcommand{\lowerint}[1]{L_{#1}}
\newcommand{\rsum}[1]{{\textstyle\sum_{#1}}}
% extra auxiliary and additional topic/proof
\newcommand{\extopic}{\bigstar}
\newcommand{\auxtopic}{\blacklozenge}
\newcommand{\additional}{\oplus}
\newcommand{\partitions}[1]{\mathcal{P}_{#1}}
\newcommand{\sampleset}[1]{\mathcal{S}_{#1}}
\newcommand{\erf}{\operatorname{erf}}
$$
Proposition 32.1 (Integrating Bounded Functions) The integral is continuous
Theorem 32.1 (The Fundamental Theorem of Calculus I)
- Proof from the axioms
- Proof from the Riemann Integral
- Proof from the Darboux Integral (Exercise)
Theorem 32.2 (The Fundamental Theorem of Calculus II)
Definition 32.1 (Endpoint Evaluation) \[f|_{[a,b]} = f(b)-f(a)\]