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 $).

Comments

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