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University of Illinois at Urbana-Champaign Fall 2006 Math 444 Group E13 Final Exam. Wednesday, December 13. 3 hours. You are allowed to use your textbook, but no other kind of documentation. Calculators, mobile phones and other electronic devices are prohibited. NAME SIGNATURE

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α−1 α−1(u ) n u lim n un n n
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sup{f(x): x∈ (a,b)}≤ sup{f(x): x∈ [a,b]}
f(a) = sup{f(x): x∈ [a,b]} f(a) = sup{f(x): x∈ (a,b)}
f(b) = sup{f(x): x∈ [a,b]}
sup{f(x): x∈ (a,b)} = sup{f(x): x∈ [a,b]}
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f(0) = 0
f 0 a∈R f(x) =ax x∈R
nf(λ x) 0a =f (0)
nλ x
f 0
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f: [0,1]→ [0,1] x∈ [0,1]
f(x) =x
E ={x∈ [0,1]: f(x)>x} 0∈E x = sup(E)
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X P(X) X
P(X) ={A: A⊂X} X,Y f: X→Y
ˆ ˆf:P(X)→P(Y) f(A) =f(A) A⊂X
ˆf f
−1˜ ˜ ˜f:P(Y)→P(X) f(B) =f (B) B⊂Y f(∅)
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X A⊂B B⊂A x∈X
x∈A⇒x∈B x∈B⇒x∈A
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