# General definition of growth in mathematics

From high school math one knows "linear growth", "exponential growth", "logistic growth", "bounded growth" etc., but is there a common accepted general definition of "growth" which covers the special cases above? Perhaps "growth" is just the same as a strictly monotonically increasing function ($\mathbb{R} \to \mathbb{R}$).

However for example the term "exponential growth" often covers also the case of "exponential decay" (as mentioned in the wikipedia article: http://en.wikipedia.org/wiki/Exponential_growth). So one might say that "growth" in general means just a strictly montonic function.

So is there a common accepted and used definition of the general concept "growth" in mathematics. If so, do you have references?

Edit: Just found a similar definition of "growth" in the german wikipedia (growth = Wachstum in german): http://de.wikipedia.org/wiki/Wachstum_%28Mathematik%29

In the section "Mathematische Beschreibung" growth is described as the behaviour of a measured quantity in time. One first determines the value of this quantity $W_1$ at time $t_1$ than at time $t_2$ ($t_2$ > $t_1$) $W_2$. If $W_2 > W_1$ the growth is called positive growth. If $W_1 < W_2$ negative growth and if $W_1 = W_2$ zero-growth.

However there are no references in this article and it is not clear whether $W_2 > W_1$ must hold for all $t_2 > t_1$ or just for one pair $t_2 > t_1$. And it's not clear to me if this definition (if made more rigorous) is commonly used.

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I don't think there's an actual definition, it's just an English term used as such. If you want, you can picture the set of all functions with a given amount of regularity as being partitioned according to asymptotic relations like $\sim$ or $=\Theta(\cdot)$, and "growth" can refer to a particular equivalence class named after a prototypical representative element. | Why was this question downvoted? – anon Apr 22 '12 at 8:32
– dtldarek Apr 22 '12 at 8:45

## 1 Answer

I agree: there is no rigorous definition of growth. In modern mathematics, growth is something relative and not absolute. So you speak of exponential growth for the function $u$ if $$\lim_{x \to +\infty} \frac{u(x)}{e^{px}}=\ell\neq 0$$ for some $p>0$, or $$0<\limsup_{x \to +\infty} \left| \frac{u(x)}{e^{px}}\right|<+\infty.$$ Similarly, you speak of polynomial growth. In other words, growth understands some comparison class of functions.

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