#Marks for some questions

Contents
    Mathematical Analysis

1.[2014-08-29] $f$ is uniformly continuous on $[a,b]$, and $int_a^infty {fleft( t ight)dt} $ is convergent. Prove that $fleft( x ight) o 0$ as $x o 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. $fleft( c ight) = 0$, prove that there exists $zeta in left( {a,b} ight)$ s.t. $fleft( zeta ight) = frac{ {fleft( zeta ight) - fleft( a ight)}}{ {b - a}}$.

3.[2014-09-20] Suppose $f$ is continuous in $(a, infty )$ and $mathop {lim }limits_{x o infty } sin fleft( x ight) = 1$. Prove that $mathop {lim }limits_{x o infty } fleft( x ight)$ exisits.

    Real Analysis

1.[2014-08-12] $g$ is a real function on a closed interval $left[ {a,b} ight]$ and $c le gleft( x ight) le d$ where $c,d e pm infty $. Let $H = left{ {x in left( {a,b} ight):gleft( x ight){ m{ ~exists~ and ~}}gleft( x ight) e 0} ight}$. If $E subseteq left[ {c,d} ight]$ and $mleft( E ight) = 0$ where $m$ is Lebesgue measure, then does $mleft( { {g^{ - 1}}left( E ight) cap H} ight) = 0$? How about $c = - infty $ & $d = infty $?

2.[2014-08-13] If $f$ is integrable, then the set $Nleft( f ight) = left{ {x:fleft( x ight) e 0} ight}$ is $sigma $-finite.

3.[2014-08-16] If $fleft( t ight)$ is Lebesgue-integrable over $left( { - infty , + infty } ight)$ and if $ - infty < a < b < infty $, then for any real nubmer $h$,[int_{left[ {a,b} ight]} {fleft( {x + h} ight)dx} = int_{left[ {a + h,b + h} ight]} {fleft( x ight)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.

    Functional Analysis
    Matrix Analysis
Contents Mathematical Analysis 1.[2014-08-29] $f$ is uniformly continuous on $[a,b]$, and $int_a^infty {fleft( t ight)dt} $ is convergent. Prove that $fleft( x ight) o 0$ as $x o 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. $fleft( c ight) = 0$, prove that there exists $zeta in left( {a,b} ight)$ s.t. $fleft( zeta ight) = frac{ {fleft( zeta ight) - fleft( a ight)}}{ {b - a}}$. 3.[2014-09-20] Suppose $f$ is continuous in $(a, infty )$ and $mathop {lim }limits_{x o infty } sin fleft( x ight) = 1$. Prove that $mathop {lim }limits_{x o infty } fleft( x ight)$ exisits. Real Analysis 1.[2014-08-12] $g$ is a real function on a closed interval $left[ {a,b} ight]$ and $c le gleft( x ight) le d$ where $c,d e pm infty $. Let $H = left{ {x in left( {a,b} ight):gleft( x ight){ m{ ~exists~ and ~}}gleft( x ight) e 0} ight}$. If $E subseteq left[ {c,d} ight]$ and $mleft( E ight) = 0$ where $m$ is Lebesgue measure, then does $mleft( { {g^{ - 1}}left( E ight) cap H} ight) = 0$? How about $c = - infty $ & $d = infty $? 2.[2014-08-13] If $f$ is integrable, then the set $Nleft( f ight) = left{ {x:fleft( x ight) e 0} ight}$ is $sigma $-finite. 3.[2014-08-16] If $fleft( t ight)$ is Lebesgue-integrable over $left( { - infty , + infty } ight)$ and if $ - infty < a < b < infty $, then for any real nubmer $h$,[int_{left[ {a,b} ight]} {fleft( {x + h} ight)dx} = int_{left[ {a + h,b + h} ight]} {fleft( x ight)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. Functional Analysis Matrix Analysis
经验分享 程序员 微信小程序 职场和发展