无穷远处为常数的函数，直觉上导数应该是零。但这个事实只对有充分高的正则性的函数才成立，试看反例和证明。
%5Cin%20C%5E%7B1%7D#card=math&code=f%28x%29%5Cin%20C%5E%7B1%7D&height=23&width=73)，且#card=math&code=f%28x%29&height=20&width=32)单增，
%3DC#card=math&code=%5Clim%5Climits_%7Bx%5Clongrightarrow%20%5Cinfty%7Df%28x%29%3DC&height=28&width=107)，不一定有
%5Clongrightarrow%200#card=math&code=f%27%28x%29%5Clongrightarrow%200&height=21&width=82),但%5Cin%20C%5E%7B1%2C%5Calpha%7D#card=math&code=f%28x%29%5Cin%20C%5E%7B1%2C%5Calpha%7D&height=23&width=84)则可以得到结论。
反例 考虑%E2%80%8B#card=math&code=f%27%28x%29%E2%80%8B&height=21&width=37)，在
上取值为
%26%3D1%20%5C%5C%0A%20%20%20%20%20%20%20%20%20f’(x)%26%3Dn%5E%7B2%7Dx%2B1-n%5E%7B3%7D-%5Cfrac%7Bn%5E%7B2%7D%7D%7B2%7D%5Cquad%20in%5Cquad%20x%5Cin%20%5Bn%2B%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7Bn%5E%7B2%7D%7D%2Cn%2B%5Cfrac%7B1%7D%7B2%7D%5D%5C%5C%0A%20%20%20%20%20%20%20%20%20f’(x)%26%3D-n%5E%7B2%7Dx%2B1%2Bn%5E%7B3%7D%2B%5Cfrac%7Bn%5E%7B2%7D%7D%7B2%7D%5Cquad%20in%5Cquad%20x%5Cin%20%5Bn%2B%5Cfrac%7B1%7D%7B2%7D%2Cn%2B%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B1%7D%7Bn%5E%7B2%7D%7D%5D%0A%5Cend%7Bsplit%7D%E2%80%8B#card=math&code=%5Cbegin%7Bsplit%7D%0A%20%20%20%20%20%20%20%20%20f%27%28n%2B%5Cfrac%7B1%7D%7B2%7D%29%26%3D1%20%5C%5C%0A%20%20%20%20%20%20%20%20%20f%27%28x%29%26%3Dn%5E%7B2%7Dx%2B1-n%5E%7B3%7D-%5Cfrac%7Bn%5E%7B2%7D%7D%7B2%7D%5Cquad%20in%5Cquad%20x%5Cin%20%5Bn%2B%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7Bn%5E%7B2%7D%7D%2Cn%2B%5Cfrac%7B1%7D%7B2%7D%5D%5C%5C%0A%20%20%20%20%20%20%20%20%20f%27%28x%29%26%3D-n%5E%7B2%7Dx%2B1%2Bn%5E%7B3%7D%2B%5Cfrac%7Bn%5E%7B2%7D%7D%7B2%7D%5Cquad%20in%5Cquad%20x%5Cin%20%5Bn%2B%5Cfrac%7B1%7D%7B2%7D%2Cn%2B%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B1%7D%7Bn%5E%7B2%7D%7D%5D%0A%5Cend%7Bsplit%7D%E2%80%8B&height=124&width=484)
于是%3D%5Cint%5E%7Bx%7D%7B0%7Df’(t)dt%E2%80%8B#card=math&code=f%28x%29%3D%5Cint%5E%7Bx%7D%7B0%7Df%27%28t%29dt%E2%80%8B&height=42&width=131)是单增的，且
%E2%80%8B#card=math&code=f%27%28t%29%E2%80%8B&height=21&width=33)连续，故%5Cin%20C%5E%7B1%7D%E2%80%8B#card=math&code=f%28x%29%5Cin%20C%5E%7B1%7D%E2%80%8B&height=23&width=73),且
%3D1%2B%5Cfrac%7B1%7D%7B2%5E%7B2%7D%7D%2B%5Cfrac%7B1%7D%7B3%5E%7B2%7D%7D%2B%5Ccdots%3C%5Cinfty%E2%80%8B%0A#card=math&code=%5Clim%5Climits_%7Bx%5Clongleftarrow%20%5Cinfty%7Df%28x%29%3D1%2B%5Cfrac%7B1%7D%7B2%5E%7B2%7D%7D%2B%5Cfrac%7B1%7D%7B3%5E%7B2%7D%7D%2B%5Ccdots%3C%5Cinfty%E2%80%8B%0A&height=41&width=268)
单增的有界数列必有极限。而%E2%80%8B#card=math&code=f%27%28x%29%E2%80%8B&height=21&width=37)显然不趋于0.
证明
若%5Cin%20C%5E%7B1%2C%5Calpha%7D#card=math&code=f%28x%29%5Cin%20C%5E%7B1%2C%5Calpha%7D&height=23&width=84),则%5Cin%20C%5E%7B%5Calpha%7D#card=math&code=f%27%28x%29%5Cin%20C%5E%7B%5Calpha%7D&height=21&width=80),若%5Cnrightarrow%200%2C%5Cforall%20%5Cvarepsilon%3E0%2C#card=math&code=f%28x%29%5Cnrightarrow%200%2C%5Cforall%20%5Cvarepsilon%3E0%2C&height=20&width=127)有
%3E%5Cvarepsilon#card=math&code=x%7Bn%7D%5Clongrightarrow%20%5Cinfty%20%2Cf%27%28x%7Bn%7D%29%3E%5Cvarepsilon&height=21&width=156).
在
上，有
dt%3Df(x%7Bn%2B1%7D)-f(x%7Bn-1%7D)%5Clongrightarrow%200.#card=math&code=%5Cint%7Bx%7Bn%7D-1%7D%5E%7Bx%7Bn%7D%2B1%7Df%27%28t%29dt%3Df%28x%7Bn%2B1%7D%29-f%28x_%7Bn-1%7D%29%5Clongrightarrow%200.&height=47&width=304)
又由%5Cin%20C%5E%7B%5Calpha%7D#card=math&code=f%27%28x%29%5Cin%20C%5E%7B%5Calpha%7D&height=21&width=80)，有
-f’(y)%5Crvert%5Cleq%20L%5Clvert%20x-y%5Crvert%5E%7B%5Calpha%7D%20%2C#card=math&code=%5Clvert%20f%27%28x%29-f%27%28y%29%5Crvert%5Cleq%20L%5Clvert%20x-y%5Crvert%5E%7B%5Calpha%7D%20%2C&height=21&width=198)当
时，-f’(y)%5Crvert%3C%5Cfrac%7B%5Cvarepsilon%7D%7B2%7D.#card=math&code=%5Clvert%20f%27%28x%29-f%27%28y%29%5Crvert%3C%5Cfrac%7B%5Cvarepsilon%7D%7B2%7D.&height=33&width=144)
于是对
%5E%7B%5Cfrac%7B1%7D%7B%5Calpha%7D%7D%2Cx%7Bn%7D%5D%2C%20f’(x)%3E%5Cfrac%7B%5Cvarepsilon%7D%7B2%7D.#card=math&code=x%5Cin%20%5Bx%7Bn%7D-%28%5Cfrac%7B%5Cvarepsilon%7D%7B2L%7D%29%5E%7B%5Cfrac%7B1%7D%7B%5Calpha%7D%7D%2Cx_%7Bn%7D%5D%2C%20f%27%28x%29%3E%5Cfrac%7B%5Cvarepsilon%7D%7B2%7D.&height=35&width=243)故
dt%5Cgeq%20%5Cfrac%7B%5Cvarepsilon%7D%7B2%7D%5Ctimes%20(%5Cfrac%7B%5Cvarepsilon%7D%7B2L%7D)%5E%7B%5Cfrac%7B1%7D%7B%5Calpha%7D%7D%0A#card=math&code=%5Cint%7Bx%7Bn%7D-1%7D%5E%7Bx_%7Bn%7D%2B1%7Df%27%28t%29dt%5Cgeq%20%5Cfrac%7B%5Cvarepsilon%7D%7B2%7D%5Ctimes%20%28%5Cfrac%7B%5Cvarepsilon%7D%7B2L%7D%29%5E%7B%5Cfrac%7B1%7D%7B%5Calpha%7D%7D%0A&height=47&width=210)
矛盾.
