摘要
处处有导数的函数(导函数)有两个很好的性质:(1)在一点处有极限,则该点必连续,若无极限则该点两侧或单侧必振荡;(2)可能有不连续点的导函数介值定理仍成立。如果函数某点的领域内处处可导,我们可得到如下三个推论:(1)当f′(x0+0)=f′(x0-0)时,则存在且连续。(2)当f′(x0+0)≠f′(x0-0),或至少有一个单侧极限为无穷时,函数在该点不可导,(3)当f′(x0+0)和f′(x0-0)中一个或同时振荡时,函数在该点可能可导。
The differentiable function is continuous or discontinuous with the mode of oscillation. And we get, if a function is differentiable in deleted neighborhood of x0 , the following consequences that ( 1 ) f^l (x0) exists ( or lim 0→x0 f^l(x) =f^l(x0)o), when f^l(x0 +0) =f^l(x0 -0),(2)f^l(x0) doesn'texist,whenf(x0 +0) ≠f^l(x0 -0) , either of f^l ( x0 + 0) and f^l (x0 -0) or both approach ∞ , and ( 3 ) f^l (x0 ) maybe exist, when either of f^l (x^0 + 0) and f^l (x0 -0 ) or both discontinuous with the mode of oscillation.
出处
《武警学院学报》
2009年第8期94-96,共3页
Journal of the Armed Police Academy
关键词
导函数
可导
连续
介值定理
derivative function
continuity
derivative
the intermediate value theorem