关于一致连续的专题讨论

$f{命题：}$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：$lim limits_{x o egin{array}{*{20}{c}}{{a^ + }}end{array}} fleft( x ight)$与$lim limits_{x o egin{array}{*{20}{c}}{{b^ -}}end{array}} fleft( x ight)$均存在且有限

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$f{命题：}$设$f(x)$在$left[ {a, + infty } ight)$上连续，若$lim limits_{x o egin{array}{*{20}{c}}{ + infty }end{array}} fleft( x ight)$存在且有限，则$f(x)$在$left[ {a, + infty } ight)$上一致连续

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$f{命题：}$设$f(x)$在$left( {a,b} ight)$上可导且导函数有界，则$f(x)$在$left( {a,b} ight)$上一致连续

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$f{命题：}$(1)设$f(x)$在$left[ {a, + infty } ight)$上可导，且$lim limits_{x o egin{array}{*{20}{c}}{ + infty }end{array}} f'left( x ight) = A $，则$f(x)$在$left[ {a, + infty } ight)$上一致连续

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(2)设$f(x)$在$left[ {a, + infty } ight)$上可导，且$lim limits_{x o egin{array}{*{20}{c}}{ + infty }end{array}} f'left( x ight) ={ + infty }$，则$f(x)$在$left[ {a, + infty } ight)$上非一致连续

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$f{命题：}$设$f(x)$在有限区间$left( {a,b} ight)$上可导，且$lim limits_{x o egin{array}{*{20}{c}}{{a^ + }}end{array}} f'left( x ight)$与$ limlimits_{x o egin{array}{*{20}{c}}{{b^ - }}end{array}} f'left( x ight)$均存在，则$f(x)$在$left( {a,b} ight)$上一致连续

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$f{命题：}$设$f(x)$定义在开区间$(a,b)$上，若对任意的$cin (a,b)$都有$lim limits_{x o c} fleft( x ight)$存在，且$lim limits_{x o egin{array}{*{20}{c}}{{a^ + }}end{array}} fleft( x ight)$与$ limlimits_{x o egin{array}{*{20}{c}}{{b^ - }}end{array}} fleft( x ight)$也存在，则$f(x)$在$(a,b)$上有界

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$f{命题：}$设$f(x)$在$<a,b>$上一致连续，$forall x in  < a,b > ,fleft( x ight) in  < c,d > $，且$g(y)$在$<c,d>$上一致连续，则$Fleft( x ight) = gleft( {fleft( x ight)} ight)$在$<a,b>$上一致连续

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$f{命题：}$设$f(x)$在$left[ {a, + infty } ight)$上一致连续，$varphi left( x ight)$在$left[ {a, + infty } ight)$上连续，且$lim limits_{x o  + infty } left[ {fleft( x ight) - varphi left( x ight)} ight] = 0$，则$varphi left( x ight)$在$left[ {a, + infty } ight)$上一致连续

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$f{命题：}$设$f(x)$在$R$上连续，且$lim limits_{x o infty } fleft( x ight)$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

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$f{命题：}$设$f(x)$在$(0,+ infty )$上可导，且$sqrt x f'left( x ight)$在$(0,+ infty )$上有界，则

   (1)$f(x)$在$(0,+ infty )$一致连续

   (2)$fleft( {{0^ + }} ight) = lim limits_{x o egin{array}{*{20}{c}}{{0^ + }}end{array}} fleft( x ight)$

   (3)若将条件“$sqrt x f'left( x ight)$在$(0,+ infty )$上有界”改为“$lim limits_{x o egin{array}{*{20}{c}}{{0^ + }}end{array}} sqrt x f'left( x ight)$和$lim limits_{x o  + infty } sqrt x f'left( x ight)$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

$f{命题：}$(1)证明：$fleft( x ight) = sqrt x $在$(0,+ infty )$上一致连续

(2)讨论$fleft( x ight) = sqrt x $在$(0,+ infty )$上是否$f{Lipschitz}$连续，即是否存在常数$L>0$，使得[left| {fleft( x ight) - fleft( y ight)} ight| le Lleft| {x - y} ight|,forall x,y in (0, + infty )]

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$f{命题：}$设$fleft( x ight) = {x^alpha }$，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

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$f{命题：}$设$f(x)$在$left[ {a, + infty } ight)left( {a > 0} ight)$上满足$f{Lipschitz}$条件，即存在$M>0$，使得对任意的$x,yin left[ {a, + infty } ight) $，有[left| {fleft( x ight) - fleft( y ight)} ight| le Mleft| {x - y} ight|]则$frac{{fleft( x ight)}}{x}$在$left[ {a, + infty } ight)$上一致连续

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$f{命题：}$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$left{ {{x_n}} ight} subset I$，$left{ {fleft( {{x_n}} ight)} ight}$也是$Cauchy$列

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$f{命题：}$

$f{命题：}$设$f(x)$在$left( { - infty , + infty } ight)$上一致连续，则存在非负实数$A,B$，使得对任意$x in left( { - infty , + infty } ight)$，有$left| {fleft( x ight)} ight| le Aleft| x ight| + B$

参考答案