关于连续性的专题讨论

$f命题：$设$f(x)in C(a,b)$，且在有理点处$(稠密子集上)$取值为$A$，则$fleft( x ight) equiv A$

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$f命题：$设$f(x)in C[a,b]$，且对任意$xin [a,b]$，存在$yin [a,b]$，使得$left| {fleft( y ight)} ight| le frac{1}{2}left| {fleft( x ight)} ight|$，则$f(x)$在$[a,b]$中有零点

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$f命题：$设$fleft( x ight) in Cleft[ {0,1} ight]$，且$fleft( 0 ight) = fleft( 1 ight)$

   (1)证明：存在${x_0} in left[ {0,frac{1}{2}} ight]$，使得$fleft( {{x_0}} ight) = fleft( {{x_0} + frac{1}{2}} ight)$

   (2)试推测：对任何的自然数$n$，是否存在${x_0} in [0,frac{{n - 1}}{n}]$，使得$fleft( {{x_0}} ight) = fleft( {{x_0} + frac{1}{n}} ight)$，并证明你的结论

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$f命题：$设$f(x)$在$x=0$处连续，且$lim limits_{x o egin{array}{*{20}{c}}0end{array}} frac{{fleft( {2x} ight) - fleft( x ight)}}{x} = A$，证明：$f'left( 0 ight) = A$

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$f命题：$设$f(x)$在$left( { - infty , + infty } ight)$上连续，若$lim limits_{x o egin{array}{*{20}{c}}infty end{array}} fleft( x ight) = + infty $，证明：

   (1)$f(x)$在$left( { - infty , + infty } ight)$上有最小值$a$

   (2)若$f(a)>a$，则$f(f(x))$在$left( { - infty , + infty } ight)$上至少两点取到最小值

$f命题：$设$f(x)$在$[a,b]$上具有介值性，且$(a,b)$内可导，$left| {f'left( x ight)} ight| leqslant k,xin (a,b)$，证明：$f(x)$在$a$处右连续，在$b$处左连续

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$f命题：$

$(04大连理工)$设$f(x)$在$[a,b]$上连续，对$xin[a,b]$，定义$mleft( x ight) = mathop {inf }limits_{a leqslant t leqslant x} fleft( t ight)$，证明：$m(x)$在$[a,b]$上连续