证明函数的连续性.

A . A、黎曼 B . B、魏尔斯特拉斯 C . C、柯西 D . D、以上均不是

证明:(1)函数<img src='https://img2.soutiyun.com/ask/2021-01-10/979120252580493.png' />在原点(0,0)连续,但不存在偏导数<img src='https://img2.soutiyun.com/ask/2021-01-10/979120271103552.png' />. (2)函数<img src='https://img2.soutiyun.com/ask/2021-01-10/979120281243132.png' />在原点(0,0)不连续[见题1(3)],但有偏导数<img src='https://img2.soutiyun.com/ask/2021-01-10/979120383280969.png' />.

证明:若函数f(x)在[a,b]连续、非负,且<img src='https://img2.soutiyun.com/ask/2020-11-12/97406224084526.png' />使f(x0)＞0,则 <img src='https://img2.soutiyun.com/ask/2020-11-12/974062250390806.png' />

证明:若函数f(x)在a连续,则函数 <img src='https://img2.soutiyun.com/ask/2020-11-11/973957882099598.png' /> 在a都连续.

证明:若函数f(x)在[O,+∞)连续,且<img src='https://img2.soutiyun.com/ask/2020-11-12/974063888799517.png' />则<img src='https://img2.soutiyun.com/ask/2020-11-12/974063900815204.png' />

证明函数<img src='https://img2.soutiyun.com/ask/2021-01-25/980442368495369.png' />连续，并有连续各阶段导函数。

设f(x)是以T为周期的连续周期函数(-∞＜x＜+∞).证明: <img src='https://img2.soutiyun.com/ask/2020-12-13/976722698275577.png' />

设ƒ (χ)在(-∞, +∞)内连续，且ƒ (χ)＞0.证明函数<img src='https://img2.soutiyun.com/ask/2020-08-30/967622834839226.png' />在(0,+ ∞)内为单调增加函数.

证明:若函数f(x)与φ(x)在[a,b]连续,则 <img src='https://img2.soutiyun.com/ask/2020-11-12/974060778329609.png' />

设函数f(u,v)在R²上具有二阶连续偏导数。证明:函数 <img src='https://img2.soutiyun.com/ask/2021-02-02/981107588858527.png' />

证明:若函数f(x)在(a,+∞)连续,且<img src='https://img2.soutiyun.com/ask/2020-11-11/97395757022676.png' />则f(x)在(a,+∞)有界.

证明:函数<img src='https://img2.soutiyun.com/ask/2021-01-10/979120577570081.png' />在点(0,0)连续且有偏导数,但不可微分.

证明:函数<img src='https://img2.soutiyun.com/ask/2020-11-13/974118519763706.png' />在[0,+∞]连续.

证明:T函数在区间(0,+∞)存在任意阶连续导数,n∈N+,有 <img src='https://img2.soutiyun.com/ask/2020-11-13/974144501001748.png' />

设函数<img src='https://img2.soutiyun.com/ask/2020-12-16/976976603992918.png' />,其中函数g(x)在(-∞,+∞)上连续,且 g(1)=5,<img src='https://img2.soutiyun.com/ask/2020-12-16/976976616554637.png' />,证明<img src='https://img2.soutiyun.com/ask/2020-12-16/976976676821084.png' />,并计算f''(1)和F'''(1).

证明：函数<img src='https://img2.soutiyun.com/ask/2021-01-28/980690877247974.png' />上连续，且有各阶连续导函数;函数<img src='https://img2.soutiyun.com/ask/2021-01-28/980690897786149.png' />上连续，且有各阶连续导函数。

设f是有界开区域<img src='https://img2.soutiyun.com/ask/2021-01-28/98070296771451.png' />上的一致连续函数。证明： <img src='https://img2.soutiyun.com/ask/2021-01-28/980702977012042.png' />

证明曲面<img src='https://img2.soutiyun.com/ask/2021-01-29/980784359816037.png' />在任一点处的切平面都通过原点，其中函数f连续可微。

设一元函数f(u)在[-1,1]上连续，证明 <img src='https://img2.soutiyun.com/ask/2021-02-01/981026749760827.png' /> 其中Ω为单位球<img src='https://img2.soutiyun.com/ask/2021-02-01/981026759890406.png' />。