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Definition
The following are equivalent:
1)X is countable
2)There is a surjection ℕ->X
3) There is an injection X->ℕ |
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Definition
Let S be nonempty.
1) An Upperbound for S is u∈ℝ s.t. s≤u ∀s∈S
If S has an upperbound, say S is bounded above.
2) A lowerbound for S is l∈ℝ s.t. l≤s ∀s∈S
If S has a lower bound, say S is bounded below
S is bounded if bounded above and below |
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Definition
Let S be nonempty.
Say x is a supremum (least upper bound) if
1)x is an upperbound
2)if u is another upperbound, x≤u
Say y is an infimum (greatest lower bound) if
1)y is a lower bound
2)if l is another lowerbound, y≥l |
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Definition
Let S be nonempty. Then u=supS iff
1)u≥s s∈S
2)If v less than u then s∈S s.t. s>v |
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Definition
Let S be nonempty.
Then u=supS⇔∀ε>0 there is some s∈S s.t. s>u-ε |
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Definition
| If ∅≠S⊆ℝ and S is bounded above the supS exists in ℝ. |
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| Thm: Nested Intervals Property |
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Definition
| Let I1⊇I2⊇I3... be nested, nonempty, closed intervals.
Then
1)∩nIn≠∅
2) If In=[an,bn] and infn{bn-an}=0, Then ∩nIn contains a single point |
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Definition
| A seq (xn) converges to a∈ℝ if for every ε>0, there is Kε∈ℕ s.t. ∀n>Kε, |xn-a|<ε
We say seq converges if it converges to some a.
If seq doesnt converge to any a, then it diverges
Alt: lim(n→oo) xna |
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Definition
| If (xn) converges, then it is bounded. |
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Definition
| Let xn>0 ∀n
Suppose lim xn+1/xn=L<1
Then xn→0 |
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Definition
Say (Xn) is Cauchy if for every E>0 there exists k s.t. for every n,m >/k then
|Xn-Xm| |
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Definition
Let r be in R.
Xn=r^n
Sn=1+r+r^2+... = [1-r^(n+1)]/[1-r]
If |r|<1 then r^n->0 so series converges to 1/(1-r)
If |r|>1 then r^n diverges
If r=-1 then (Sn) is (1,0,1,0,1,...) diverges
r=/=1 |
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| Thm: Cluster pt and convergence |
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Definition
| c is a cluster point of a set A iff there is a seq (Xn) in A converging to c |
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Definition
| Let f:A->R and c be a cluster pt of A.
Say L is the limit of f at c if for every E>0, there exists a Delta s.t. x in VsubDelta(c)IntersectionA\{c}, then f(x) is in VsubE(c).
For x in A, c-Deltac of f(x) may exist even if f(x) is not defined at x=c
-limits are unique when they exist |
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| Thm: Sequential Criterion for Limits |
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Definition
Let A be contained in F, f:A->R, c is a cluster pt.
Then limx->c of f(x)=L iff for EVERY seq (Xn) in A\{c} that converges to c, then f(Xn)->L. |
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Definition
Let A be contained in R and f:A->R and c in A. Say f is continuous at c if either
1) c is a cluster pt of A and limx->c of f(x) exists and is equal to f(c)
2) c is not a cluster pt
By def, f is automatically continous at any isolated pt.
f is Discontinuous at c if not cont at c.
f is cont on A if cont at all c in A |
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| Thm: Equivalent Statements of cont f |
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Definition
| Let A be contained in R, f:A->R a function, c in A. Then the following are equivalent.
1)f is cont at c
2)For every E>0, there exists Delta>0 s.t. if |x-c|0, there exists Delta>0 s.t. if x is in VsubE(c)IntersectionA then f(x) is in VsubE(f(c))
<-> f(VsubDelta(c)IntersectionA) is contained in VsubE(f(c))
<->VsubDelta(c)IntersectionA contained in f^-1(VsubE(f(c)) |
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| Thm: Continuous Functions on (closed, bounded) intervals |
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Definition
Let f:I->R be cont where I is a closed and bdd interval. Then f is bdd.
Note: Result if false if I is not closed or bdd |
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Definition
Let f:I=[a,b]->R be cont.
Then there exists a pt c in I s.t. f(c)=supf(I)
There exists some point d in I s.t. f(d)=inff(I)
Say that f attains the max on I |
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| Thm: Intermediate Value Theorem |
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Definition
| Let f:I->R be cont. Suppose a,b are in f(I) and f(a)Then there exists a pt c in I s.t. f(c)=z. |
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| Def: Uniformly Continuous |
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Definition
| We say that f:A->R is uniformly cont on A if for every E>0, there exists Delta>0 s.t. if |x-y| |
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| Thm: Uniform Continuity Theorem |
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Definition
| If f is cont on [a,b], then it is uniformly cont on [a,b] |
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| Thm: Continuous Extension Theorem |
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Definition
Let f be cont on (a,b). Then f is uniform cont on (a,b) iff there exists a continuous extension g(x) on [a,b]
(If we restrict g(x) to (a,b), it is f(x)) |
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| Lemma: Uniform and Cauchy |
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Definition
| If f is uniform cont on A and (Xn) is Cauchy in A, then f(Xn) is Cauchy |
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Definition
f:J->ℝ is diff at c∈J iff ∃φ:I->ℝ cont at c and f(x)-f(c)=φ(x)(x-c).
Then f'(c)=φ(c) |
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| Thm: Diff and Inverse functions |
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Definition
Let I∈ℝ and let f:I->ℝ be strictly monotone and cont on I.
If f is diff at c on I and f'(c)≠0, then f-1 is diff at d:=f(c) and (f-1)'(d)=1/[f'(f-1(d))] |
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Definition
| Let f:[a,b]->ℝ be cont and diff on (a,b). Then ∃c∈(a,b) s.t. f'(c)= [f(b)-f(a)]/[b-a] |
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Definition
Given f:I->ℝ and a tagged partition P., define the Riemann Sum of f (using P) to be S(f,P.)= ∑(from i=1 to n) f(ti)(xi-x(i-1))
Where f(ti) is the height of the rect. |
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Definition
Say that f:I->ℝ is (Riemann) integrable if ∃L∈ℝ and ∀ε>0, ∃δ>0 such that ∀P satisfying ||P||<δ, then |S(f,P)-L|<ε
Think of this as saying
lim||P||->0 of S(f,P)=L
If so, write ∫a to b f(x)dx=L
Riemann Integrable = R |
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Definition
| If f is unbdd on [a,b], then f∉R[a,b] |
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| Thm: Cauchy Criterion for Integration |
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Definition
f:[a,b]->ℝ iff
∀ε>0, ∃γ s.t. if ||P||<γ and ||Q||<γ then |S(f,P)-S(f,Q)|<ε |
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Definition
| Let f:[a,b]->ℝ. Say F:[a,b]->ℝ is an anti derivative (primitive) for f if F'(x)=f(x) |
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Definition
if f∈R[a,b] and F is a primitive for f on [a,b].
Then ∫a to b f(x)dx=F(b)-F(a) |
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Definition
| If f is cont on [a,b] then F(z) is diff on [a,b] AND F'(z)=f(z) |
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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 |
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Definition
| ∀a∈A, ∀ε<0, ∃N s.t. n≤N, |fn(a)-f(a)|<ε
Note: N depends on a |
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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. |
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Definition
| Let fn:A→ℝ be cont. Suppose fn→f uniformly. Then f is also cont. |
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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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