1. (15 points) Let S be a set. Show that the cardinality of the power set P ( S ) is strictly bigger than S.

HONORED ADVANCED CALCULUS MID-TERM EXAM

9:10 – 12:40, 11/08, 2011

A COURSE BY CHIN-LUNG WANG

1. (15 points) Let S be a set. Show that the cardinality of the power set P ( S ) is strictly bigger than S.

2. (15 points) Let ( S, d ) be a metric space. Show that ( S, d ) is compact ⇐⇒ ( S, d ) is sequentially compact. (Do “ ⇒ ” first. For “ ⇐ ”, first show that ( S, d ) is separable.) 3. (15 points) Let α ∈ BV [ a, b ] . Show that V ( x ) : = V

( a, x ) ∈ BV [ a, b ] and α is continu-

ous at x if and only if V is continuous at x. Finally show that if α ∈ C [ a, b ] ∩ BV [ a, b ] then α can be written as the difference of two strictly monotone continuous functions.

4. (15 points) Let α ∈ BV [ a, b ] . Show that f ∈ R ( α ) ⇒ f ∈ R ( V ) . Based on this, show that f , g ∈ R ( α ) ⇒ f g ∈ R ( α ) . Moreover, show that G ( x ) : = R x

a g dα ∈ BV [ a, b ] and f ∈ R ( G ) with R b

a f dG = R b a f g dα.

5. (15 points) Show that f ∈ R ( α ) on [ a, b ] if and only if it satisfies the Cauchy criterion:

For any e > 0, there exists P

∈ _P [ a, b ] such that | S ( P, f , α ) − S ( P ⁰ , f , α )| < e for all P, P ⁰ ⊃ P

. Use this to show that f ∈ R ( α ) on [ a, b ] ⇒ f ∈ R ( α ) on any [ c, d ] ⊂ [ a, b ] . 6. (15 points) Let ∑ ^∞ n = 0 a _n → A absolutely and ∑ ^∞ n = 0 b _n → B. Show that ∑ ^∞ n = 0 c _n → AB

where c n = _∑ ⁿ _{k = 0} a _k b _{n − k} is the Cauchy product.

7. Let f be defined and bounded on [ a, b ] _{. If T} ⊂ [ a, b ] , we define the oscillation of f on T as Ω f ( T ) = sup { f ( x ) − f ( y ) : x, y ∈ T } . The oscillation of f at x is defined to be the number ω _f ( x ) = lim _h → 0

Ω f ( B ( x, h ) ∩ [ a, b ]) .

(a) (5 points) Let ε > 0 be given. Assume that ω _f ( x ) < ε for all x ∈ [ a, b ] , show that there exists a δ > 0 (depending only on ε) such that for every closed subinterval T ⊂ [ a, b ] _{, we have Ω f} ( T ) < ε whenever the length of T is less than δ.

(b) (5 points) For any ε > 0, show that the set J

= { x ∈ [ a, b ] : ω _f ( x ) ≥ ε } is compact.

(c) (Bonus problem with 10 points) Let D denote the set of discontinuity of f in [ a, b ] . Prove that f is Riemann-integrable on [ a, b ] if and only if D has measure zero.

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