# #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