\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)

Arithmetic operations on functions

Given two functions \( f : X \to \mathbb{R} \) and [...], we can define their:

sum, \( f + g : X \to \mathbb{R} \)

\( (f + g)(x) := f(x) + g(x) \)

difference, \( f - g : X \to \mathbb{R} \)

\( (f - g)(x) := f(x) - g(x) \)

maximum, \( \max(f, g) : X \to \mathbb{R} \)

\( \max(f, g)(x) := \max(f(x), g(x)) \)

minimum, \( \min(f, g) : X \to \mathbb{R} \)

\( \min(f, g)(x) := \min(f(x), g(x)) \)

product, \( fg : X \to \mathbb{R} \) (or \( f \cdot g \) )

\( (fg)(x) := f(x)g(x) \)

quotient, assuming \( g(x) \ne 0 \text{ for all } x \in X\), \( f/g : X \to \mathbb{R} \)

\( (f/g)(x) := f(x)/g(x) \)

constant multiple, \( cf : X \to \mathbb{R} \)

\( (cf)(x) := cf(x) \)