函數之單調性者，亦省作單調性，增減性，函數連續之本德也，其況有四，增減平轉是也。
例：有函數 f ( x ) = 2 x − 1 , x ∈ [ 2 , 6 ] {\displaystyle f(x)={\frac {2}{x-1}}\ ,x\in [2,6]} 者，試論其增減性
解：設x1 , x2，且 2 ⩽ x 1 < x 2 ⩽ 6 {\displaystyle 2\leqslant x_{1}<x_{2}\leqslant 6} ，則
f ( x 2 ) − f ( x 1 ) = 2 x 2 − 1 − 2 x 1 − 1 {\displaystyle f(x_{2})-f(x_{1})={\frac {2}{x_{2}-1}}-{\frac {2}{x_{1}-1}}}
= 2 [ ( x 1 − 1 ) − ( x 2 − 1 ) ] ( x 1 − 1 ) ( x 2 − 1 ) {\displaystyle ={\frac {2[(x_{1}-1)-(x_{2}-1)]}{(x_{1}-1)(x_{2}-1)}}}
= 2 ( x 1 − x 2 ) ( x 1 − 1 ) ( x 2 − 1 ) {\displaystyle ={\frac {2(x_{1}-x_{2})}{(x_{1}-1)(x_{2}-1)}}}
= − 2 ( x 2 − x 1 ) ( x 1 − 1 ) ( x 2 − 1 ) {\displaystyle =-{\frac {2(x_{2}-x_{1})}{(x_{1}-1)(x_{2}-1)}}}
< 0 {\displaystyle <0}
故其遞減
見導數頁
![{\displaystyle f(x)={\frac {2}{x-1}}\ ,x\in [2,6]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c732a887a17f12c823c711db6b8ca1b4aad36c7b)
者，試論其增減性
，則
![{\displaystyle ={\frac {2[(x_{1}-1)-(x_{2}-1)]}{(x_{1}-1)(x_{2}-1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1eebf9d629f3cf4764088d253d5417d7ca823009)
