In my calculus book it mentions that increasing functions maintain inequality relations and that's the reason you can apply $\exp$ and $\ln$ to two sides of an inequality to solve them. Is there some general classification for the types of functions that maintain inequality? For instance are they all 1 to 1 or have some other property in common?
What functions maintain inequality?
2
$\begingroup$
algebra-precalculus
inequality
-
4Yes. A function maintains strong inequalities (<) if and only if it is increasing and weak inequalities ($\le$) if and only if it is non-decreasing. This is, in a way, the **definition** of non-decreasing. – 2012-07-09
1 Answers
3
In addition to m. k.'s answer, there is one very valuable criterion: If a function is continuous, it is monotonic if and only if it is injective (this is a consequence of the Intermediate Value Theorem). Therefore, a continuous function $f: [a,b] \to \mathbb R$ will maintain a strict inequality if and only if $f(a) < f(b)$ and it is injective.
-
0I'm testing the endpoints. The argument runs as follows:$f$maintains a strict inequality iff is is strictly monotonously increasing (SMI). SMI functions are trivially injective, and if a continuous function with f(a) < f(b) is not SMI, there are two points $x,y \in [a,b]$, x
, such that $f(x) \ge f(y)$. If $f(x) = f(y)$, $f$ is not injective, so assume f(x) > f(y). Now, we have to consider various cases, but for each case, we can apply the intermediate value theorem to find a point $z \in [a,b]$ such $z \neq z'$ for $z' \in \{ a,b,x,y\}$ and $f(z) = f(z')$. – 2012-07-16