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33 Cards in this Set
- Front
- Back
Divergence Criterion for Limits |
Let f:A->R and let c be a cluster point of A. The following are equivalent:
-IF L∈R, then f doesn not have limit L at c iff there exists a sequence xₒ in A with xₒ≠c for all n∈N s.t. the sequence xₒ converges to c but the sequence (f(xₒ)) does not converge to L. -The function f does not have a limit at c iff there exists a sequence xₒ in A with xₒ≠c for all n∈N s.t. the sequence xₒ converges to c but the sequence (f(xₒ)) does not converge in R. |
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Definition of continuous function
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Let A⊆R, let f:A->R, and let c∈A. We say that f is continuous at c if, given any number ε>0 there exists δ>0 s.t. if x is any point of A satisfying ∣x-c∣<δ, then ∣f(x)-f(c)∣<ε.
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Discontinuity Criterion
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Let A⊆R, let f:A->R, and let c∈A. Then f is discontinuous at c iff there exists a sequence (xₒ) in A s.t. (xₒ) converges to c, but the sequence f(xₒ) does not converge to f(c).
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Definition of continuous set
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Let A⊆R, let f:A->R. If B is a subset of A, we say that f is continuous on the set B if f is continuous at every point of B.
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Sequential Criterion for limits
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Let f:A->R and let c be a cluster point of A. Then the following are equivalent.
1)lim f as x approaches c = L 2)for every sequence (xₒ) in A that converges to c s.t. xₒ≠c for all n∈N, the sequence (f(xₒ)) converges to L. |
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Definition of the derivative
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Let I⊆R and f:I->R and c∈I. A real number L is the derivative of f at c if for any ε>0 there exists δ(ε)>0 s.t. if x∈I satisfies 0<∣x-c∣<δ(ε), then ∣[f(x)-f(c)/x-c]-L∣<ε.
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Caratheodory's Theorem
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Let f be defined on an interval I containing the point c. Then f is differentiable at c iff there exists a function φ on I that is continuous at c and satisfies
f(x)-f(c)=φ(x)(x-c) for x∈I. |
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Rolle's Theorem
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Suppose that f is continuous on a closed interval I:=[a,b], that derivative f' exists at every point of the open interval (a,b), and that f(a)=f(b)=0. Then there exists at least one point c in (a,b) s.t. f'(c)=0.
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Mean Value Theorem
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Suppose that f is continuous on a closed interval I:=[a,b], and that f has a derivative in the open interval (a,b). Then there exists at least one point c in (a,b) s.t. f(b)-f(a)=f'(c)(b-a).
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Definition of Cauchy Sequence
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A sequence X=(xₒ) of real numbers is a Cauchy Sequence if for every ε>0 there exists a natural number N such that for all natural numbers n,m≥N ∣x(n)-x(m)∣<ε
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Definition of Cluster Point
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Let A⊆R. A point c∈R is a cluster point of A if for every δ>0 there exists at least one point x∈A, x≠c s.t. ∣x-c∣<δ
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Definition of a limit
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Let A⊆R and let a point c∈R be a cluster point of A. for a function f:A→R, a real number L is said to be a limit of f at c if, given any ε>0 there exists δ>0 s.t. if x∈A and 0<∣x-c∣<δ, then ∣f(x)-L∣<ε
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Definition of right-handed limit
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Let A⊆R and let f:A→R.
-If c∈R is a cluster point of A⊓(c,∞)={x∈A:x>c}, then L is a right-handed limit of f at c if, given any ε>0 there exists δ=δ(ε)>0 s.t. for all x∈A with 0 |
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Definition of left-handed limit
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Let A⊆R and let f:A→R.
-If c∈R is a cluster point of A⊓(-∞,c)={x∈A:x0 there exists δ>0 s.t. for all x∈A with 0 |
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Definition of Infinite limits ∞
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Let A⊆R and let f:A→R and let c∈R be a cluster point of A.
-f tends to ∞ as x→c, if for every λ∈R there exists δ=δ(λ)>0 s.t. for all x∈A with 0<∣x-c∣<δ, then f(x)>λ |
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Definition of Infinite limits -∞
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Let A⊆R and let f:A→R and let c∈R be a cluster point of A.
-f tends to -∞ as x→c, if for every λ∈R there exists δ=δ(λ)>0 s.t. for all x∈A with 0<∣x-c∣<δ, then f(x)<λ |
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Definition of Limits at Infinity
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Let A⊆R and let f:A→R.
Suppose (a,∞)⊆A for some a∈A. Real number L is a limit of f as x→∞ if given any ε>0 there exists K(ε)>a s.t. for any x>K, then ∣f(x)-L∣<ε |
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Definition of continuous function at c
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Let A⊆R, let f:A→R, and let c∈A.
f is continuous at c if, given any ε>0 there exists δ>0 s.t. if x∈A satisfying ∣x-c∣<δ then ∣f(x)-f(c)∣<ε |
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Definition of continuous function on a set
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Let A⊆R, let f:A→R.
If B is a subset of A, f is continuous on the set B if f is continuous at every point of B. |
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Definition of uniformly continuous
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Let A⊆R, let f:A→R.
f is uniformly continuous on A if for each ε>0 there exists δ(ε)>0 s.t. if x,u∈A satisfying ∣x-c∣<δ(ε) then ∣f(x)-f(u)∣<ε |
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Definition of Lipschitz function
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Let A⊆R, let f:A→R.
If there exists a constant K>0 s.t. ∣f(x)-f(u)∣≤K∣x-u∣ for all x,u∈A then f is a Lipschitz function. |
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Open Set
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A subset G of R is open in R if for each x∈G there exists a neighborhood V of x s.t. V⊆G.
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Closed Set
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A subset G of R is closed in R if the complement C(F):=R\G is open in R.
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Definition of a partition
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A partition of an interval I:=[a,b] is a collection P={I1,....In} of non-overlapping closed intervals whose union is [a,b].
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Definition of a tagged partition
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If a point t(i) has been chosen from each Interval I(i) then the points t(i) are called the tags and the set of ordered pairs P={(I1,t1),....(I(n),t(n))) is called a tagged partition of I.
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Definition of Riemann Integral
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A function f:[a,b]→R is Riemann integrable on [a,b] if there exists a number L∈R s.t. for every ε>0 there exists δ(ε)>0 s.t. if P is any tagged partition of [a,b] with norm of P<δ(ε), then ∣S(f:P)-L∣<ε
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Definition of Riemann sum
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Define Riemann sum of a function f:[a,b]→R corresponding to a tagged partition P to be the number
S(f:P):=∑f(t(i)(x(i)-x(i-1)) |
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Cauchy Convergence Criterion
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A sequence of real numbers is convergent iff it is a Cauchy sequence.
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Limit Squeeze Theorem
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Let A⊆R, let f,g,h:A->R, and let c∈R be a cluster point of A. If
f(x)≤g(x)≤h(x) for all x∈A,x≠c and if lim f=L=lim h, then lim g=L. |
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Maximum-Minimum Theorem
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Let I:=[a,b] be a closed bounded interval and let f:I→R be continuous on I. Then f has an absolute maximum and an absolute minimum on I.
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Bolzano's Intermediate Value Theorem
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Let I be an interval and let f:I→R be continuous on I. If a,b∈I and if k∈R satisfies f(a)
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Uniform Continuity Theorem
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Let I be a closed bounded interval and let f:I∈R be continuous on I. Then f is uniformly continuous on I.
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Differentiable Implies Continuous Theorem
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If f:I→R has a derivative at c∈I, then f is continuous at c.
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