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Math 444 Ch 8
n/a
5
Mathematics
Undergraduate 3
12/16/2010
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Term
Seq of Functions
 Definition
Let {fn} be a seq of functions A→ℝ Let f:A→ℝ Say fn→f if fn(a)→f(a) ∀a∈A This is pointwise convergence
Term
ε-δ terms
 Definition
∀a∈A, ∀ε<0, ∃N s.t. n≤N, |fn(a)-f(a)|<ε Note: N depends on a
Term
Def: Uniformly
 Definition
∀ε<0, ∃N s.t. n≤N, |fn(a)-f(a)|<ε for every a in A. Now N does not depend on a.
Term
Thm: cont uniform
 Definition
Let fn:A→ℝ be cont. Suppose fn→f uniformly. Then f is also cont.
Term
Thm: uniform R
 Definition
Let fn∈R[a,b] and fn→f uniformly. Then f∈R[a,b] and ∫a to b f= lim ∫a to b fn
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