Math and science::Analysis::Tao::09. Continuous functions on R
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, \( cf : X \to \mathbb{R} \)
- \( (cf)(x) := cf(x) \)