$\bf{命题：}$设$f(x)$在$(a,b)$上连续，则$f(x)$在$(a,b)$上一致连续的充要条件是：$\lim \limits_{x \to \begin{array}{*{20}{c}}{{a^ + }}\end{array}} f\left( x \right)$与$\lim \limits_{x \to \begin{array}{*{20}{c}}{{b^ -}}\end{array}} f\left( x \right)$均存在且有限
$\bf{命题：}$设$f(x)$在$\left[ {a, + \infty } \right)$上连续，若$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}} f\left( x \right)$存在且有限，则$f(x)$在$\left[ {a, + \infty } \right)$上一致连续
$\bf{命题：}$设$f(x)$在$\left( {a,b} \right)$上可导且导函数有界，则$f(x)$在$\left( {a,b} \right)$上一致连续
$\bf{命题：}$(1)设$f(x)$在$\left[ {a, + \infty } \right)$上可导，且$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}} f'\left( x \right) = A $，则$f(x)$在$\left[ {a, + \infty } \right)$上一致连续
(2)设$f(x)$在$\left[ {a, + \infty } \right)$上可导，且$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}} f'\left( x \right) ={ + \infty }$，则$f(x)$在$\left[ {a, + \infty } \right)$上非一致连续
$\bf{命题：}$设$f(x)$在有限区间$\left( {a,b} \right)$上可导，且$\lim \limits_{x \to \begin{array}{*{20}{c}}{{a^ + }}\end{array}} f'\left( x \right)$与$ \lim\limits_{x \to \begin{array}{*{20}{c}}{{b^ - }}\end{array}} f'\left( x \right)$均存在，则$f(x)$在$\left( {a,b} \right)$上一致连续
$\bf{命题：}$设$f(x)$定义在开区间$(a,b)$上，若对任意的$c\in (a,b)$都有$\lim \limits_{x \to c} f\left( x \right)$存在，且$\lim \limits_{x \to \begin{array}{*{20}{c}}{{a^ + }}\end{array}} f\left( x \right)$与$ \lim\limits_{x \to \begin{array}{*{20}{c}}{{b^ - }}\end{array}} f\left( x \right)$也存在，则$f(x)$在$(a,b)$上有界
$\bf{命题：}$设$f(x)$在$<a,b>$上一致连续，$\forall x \in < a,b > ,f\left( x \right) \in < c,d > $，且$g(y)$在$<c,d>$上一致连续，则$F\left( x \right) = g\left( {f\left( x \right)} \right)$在$<a,b>$上一致连续
$\bf{命题：}$设$f(x)$在$\left[ {a, + \infty } \right)$上一致连续，$\varphi \left( x \right)$在$\left[ {a, + \infty } \right)$上连续，且$\lim \limits_{x \to + \infty } \left[ {f\left( x \right) - \varphi \left( x \right)} \right] = 0$，则$\varphi \left( x \right)$在$\left[ {a, + \infty } \right)$上一致连续
$\bf{命题：}$设$f(x)$在$R$上连续，且$\lim \limits_{x \to \infty } f\left( x \right)$存在，则
(1)$f(x)$在$R$上有界
(2)$f(x)$在$R$上能取得最大值或最小值
(3)$f(x)$在$R$上一致连续
$\bf{命题：}$设$f(x)$在$(0,+ \infty )$上可导，且$\sqrt x f'\left( x \right)$在$(0,+ \infty )$上有界，则
(1)$f(x)$在$(0,+ \infty )$一致连续
(2)$f\left( {{0^ + }} \right) = \lim \limits_{x \to \begin{array}{*{20}{c}}{{0^ + }}\end{array}} f\left( x \right)$
(3)若将条件“$\sqrt x f'\left( x \right)$在$(0,+ \infty )$上有界”改为“$\lim \limits_{x \to \begin{array}{*{20}{c}}{{0^ + }}\end{array}} \sqrt x f'\left( x \right)$和$\lim \limits_{x \to + \infty } \sqrt x f'\left( x \right)$都存在”，试问：还能否推出$f(x)$在$(0,+ \infty )$一致连续，如果能请给出证明，否则举出反例
参考答案
$\bf{命题：}$(1)证明：$f\left( x \right) = \sqrt x $在$(0,+ \infty )$上一致连续
(2)讨论$f\left( x \right) = \sqrt x $在$(0,+ \infty )$上是否$\bf{Lipschitz}$连续，即是否存在常数$L>0$，使得\[\left| {f\left( x \right) - f\left( y \right)} \right| \le L\left| {x - y} \right|,\forall x,y \in (0, + \infty )\]
$\bf{命题：}$设$f\left( x \right) = {x^\alpha }$，证明：当$0 < \alpha \le 1$时，$f(x)$在$(0,+ \infty )$上一致连续；当$\alpha >1$时，$f(x)$在$(0,+ \infty )$上非一致连续
$\bf{命题：}$设$f(x)$在$\left[ {a, + \infty } \right)\left( {a > 0} \right)$上满足$\bf{Lipschitz}$条件，即存在$M>0$，使得对任意的$x,y\in \left[ {a, + \infty } \right) $，有\[\left| {f\left( x \right) - f\left( y \right)} \right| \le M\left| {x - y} \right|\]则$\frac{{f\left( x \right)}}{x}$在$\left[ {a, + \infty } \right)$上一致连续
$\bf{命题：}$设$f(x)$在有限区间$I$上有定义，则$f(x)$在$I$上一致连续的充要条件是：对任意的$Cauchy$列$\left\{ {{x_n}} \right\} \subset I$，$\left\{ {f\left( {{x_n}} \right)} \right\}$也是$Cauchy$列
$\bf{命题：}$
$\bf{命题：}$设$f(x)$在$\left( { - \infty , + \infty } \right)$上一致连续，则存在非负实数$A,B$，使得对任意$x \in \left( { - \infty , + \infty } \right)$，有$\left| {f\left( x \right)} \right| \le A\left| x \right| + B$
参考答案