# Decreasing function

A function $f$ defined on a set $E$ of real numbers such that the condition

$$x ^ \prime < x ^ {\prime\prime} ,\ \ x ^ \prime , x ^ {\prime\prime} \in E,$$

implies

$$f ( x ^ \prime ) > f ( x ^ {\prime\prime} ).$$

Sometimes such a function is called strictly decreasing and the term "decreasing function" is applied to functions satisfying for the indicated values $x ^ \prime , x ^ {\prime\prime}$ only the condition $f ( x ^ \prime ) \geq f ( x ^ {\prime\prime} )$( a non-increasing function). Every strictly decreasing function has an inverse function, which is again strictly decreasing. If $x _ {0}$ is a left-hand (respectively, right-hand) limit point of $E$, $f$ is non-increasing and if the set $\{ {y } : {y = f ( x), x > x _ {0} , x \in E } \}$ is bounded from above (respectively, $\{ {y } : {y = f ( x), x < x _ {0} , x \in E } \}$ is bounded from below), then for $x \rightarrow x _ {0} + 0$( respectively, $x \rightarrow x _ {0} - 0$), $x \in E$, $f ( x)$ has a finite limit; if the given set is not bounded from above (respectively, from below), then $f ( x)$ has an infinite limit, equal to $+ \infty$( respectively, $- \infty$).

A function $f$ such that $- f$ is decreasing is called increasing (cf. Increasing function).