$\bf命题：$设$f(x)\in C(a,b)$，且在有理点处$(稠密子集上)$取值为$A$，则$f\left( x \right) \equiv A$
$\bf命题：$设$f(x)\in C[a,b]$，且对任意$x\in [a,b]$，存在$y\in [a,b]$，使得$\left| {f\left( y \right)} \right| \le \frac{1}{2}\left| {f\left( x \right)} \right|$，则$f(x)$在$[a,b]$中有零点
$\bf命题：$设$f\left( x \right) \in C\left[ {0,1} \right]$，且$f\left( 0 \right) = f\left( 1 \right)$
(1)证明：存在${x_0} \in \left[ {0,\frac{1}{2}} \right]$，使得$f\left( {{x_0}} \right) = f\left( {{x_0} + \frac{1}{2}} \right)$
(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$f\left( {{x_0}} \right) = f\left( {{x_0} + \frac{1}{n}} \right)$，并证明你的结论
$\bf命题：$设$f(x)$在$x=0$处连续，且$\lim \limits_{x \to \begin{array}{*{20}{c}}0\end{array}} \frac{{f\left( {2x} \right) - f\left( x \right)}}{x} = A$，证明：$f'\left( 0 \right) = A$
$\bf命题：$设$f(x)$在$\left( { - \infty , + \infty } \right)$上连续，若$\lim \limits_{x \to \begin{array}{*{20}{c}}\infty \end{array}} f\left( x \right) = + \infty $，证明：
(1)$f(x)$在$\left( { - \infty , + \infty } \right)$上有最小值$a$
(2)若$f(a)>a$，则$f(f(x))$在$\left( { - \infty , + \infty } \right)$上至少两点取到最小值
$\bf命题：$设$f(x)$在$[a,b]$上具有介值性，且$(a,b)$内可导，$\left| {f'\left( x \right)} \right| \leqslant k,x\in (a,b)$，证明：$f(x)$在$a$处右连续，在$b$处左连续
1
$\bf命题：$
$(04大连理工)$设$f(x)$在$[a,b]$上连续，对$x\in[a,b]$，定义$m\left( x \right) = \mathop {\inf }\limits_{a \leqslant t \leqslant x} f\left( t \right)$，证明：$m(x)$在$[a,b]$上连续