Editing Subadditivity (section)

===Functions===
{{math theorem|name='''Theorem:'''<ref>Hille 1948, Theorem 6.6.1. (Measurability is stipulated in Sect. 6.2 "Preliminaries".)</ref>|math_statement= For every [[Measurable function|measurable]] subadditive function <math>f : (0,\infty) \to \R,</math> the limit <math>\lim_{t\to\infty} \frac{f(t)}{t}</math> exists and is equal to <math>\inf_{t>0} \frac{f(t)}{t}.</math> (The limit may be <math>-\infty.</math>)}}

If ''f'' is a subadditive function, and if 0 is in its domain, then ''f''(0) ≥ 0. To see this, take the inequality at the top. <math>f(x) \ge f(x+y) - f(y)</math>. Hence <math>f(0) \ge f(0+y) - f(y) = 0</math>

A [[concave function]] <math>f: [0,\infty) \to \mathbb{R}</math> with <math>f(0) \ge 0</math> is also subadditive.
To see this, one first observes that <math>f(x) \ge \textstyle{\frac{y}{x+y}} f(0) + \textstyle{\frac{x}{x+y}} f(x+y)</math>.
Then looking at the sum of this bound for <math>f(x)</math> and <math>f(y)</math>, will finally verify that ''f'' is subadditive.<ref>{{cite book | last = Schechter | first=Eric | author-link=Eric Schechter| title=Handbook of Analysis and its Foundations | publisher=Academic Press | location = San Diego | year=1997 | isbn=978-0-12-622760-4}}, p.314,12.25</ref>

The negative of a subadditive function is [[superadditivity|superadditive]].