#Marks for some questions
Contents
1.[2014-08-29] $f$ is uniformly continuous on $[a,b]$, and $\int_a^\infty {f\left( t \right)dt} $ is convergent. Prove that $f\left( x \right) \to 0$ as $x \to \infty $.
2.[2014-09-06] Suppose $f$ is continuous on$[a,b]$ and differentiable in $(a,b)$. If there exists $c \in (a,b)$ s.t. $f'\left( c \right) = 0$, prove that there exists $\zeta \in \left( {a,b} \right)$ s.t. $f'\left( \zeta \right) = \frac{{f\left( \zeta \right) - f\left( a \right)}}{{b - a}}$.
3.[2014-09-20] Suppose $f$ is continuous in $(a, \infty )$ and $\mathop {\lim }\limits_{x \to \infty } \sin f\left( x \right) = 1$. Prove that $\mathop {\lim }\limits_{x \to \infty } f\left( x \right)$ exisits.
1.[2014-08-12] $g$ is a real function on a closed interval $\left[ {a,b} \right]$ and $c \le g\left( x \right) \le d$ where $c,d \ne \pm \infty $. Let $H = \left\{ {x \in \left( {a,b} \right):g'\left( x \right){\rm{ ~exists~ and ~}}g'\left( x \right) \ne 0} \right\}$. If $E \subseteq \left[ {c,d} \right]$ and $m\left( E \right) = 0$ where $m$ is Lebesgue measure, then does $m\left( {{g^{ - 1}}\left( E \right) \cap H} \right) = 0$? How about $c = - \infty $ & $d = \infty $?
2.[2014-08-13] If $f$ is integrable, then the set $N\left( f \right) = \left\{ {x:f\left( x \right) \ne 0} \right\}$ is $\sigma $-finite.
3.[2014-08-16] If $f\left( t \right)$ is Lebesgue-integrable over $\left( { - \infty , + \infty } \right)$ and if $ - \infty < a < b < \infty $, then for any real nubmer $h$,\[\int_{\left[ {a,b} \right]} {f\left( {x + h} \right)dx} = \int_{\left[ {a + h,b + h} \right]} {f\left( x \right)dx}. \]
4.[2014-10-18] Here are some observations regarding the set operation $A + B$.
(a) Show that if either $A$ and $B$ is open, then $A + B$ is open.
(b) Show that if $A$ and $B$ are closed, then $A + B$ is measurable.
(c) Show, however, that $A + B$ might not be closed even though $A$ and B are closed.
