NetMath 444 Midterm 3

Partition

A finite set of points in [a, b], where a = x0 < x1 < ... < xn = b.

Cauchy Criterion of Integrable

A function f : [a, b] → ℝ belongs to R[a, b] if and only if for every ε > 0 there exists η(ε) > 0 such that if P and Q are any tagged partitions of [a, b] with ||P||, ||Q|| < η(ε), then |S(f; P) - S(f; Q)| < ε.

Continuous Extension Theorem

A function f is uniformly continuous on the interval (a,b) if and only if it can be defined at the endpoints a and b s.t. the extended function is continuous on [a,b].

Equivalence Theorem

A function f on I = [a, b] is Darboux integrable if and only if it is Riemann integrable.

Step Function

A function s : [a,b] → ℝ where [a,b] is the union of a finite number of non-overlapping intervals A1, A2, ..., An s.t. s is constant on each interval, that is, s(x) = ck for ∀x ∈ Ak, where k := 1, 2, n.

Refinement

A partition Q is a refinement of P if P ⊂ Q, where U(f, Q) ≤ U(f, P) and L(f, Q) ≥ L(f, P).

Relative Extremum

If a function has a relative maximum or relative minimum. Suppose f : I → ℝ and c ∈ I. f is a {relative maximum, relative minimum} at c if ∃δ > 0 s.t. {f(c) ≥ f(x), f(c) ≤ f(x)} for ∀x ∈ Vδ(c) ∩ I.

Boundedness Theorem

If f : [a,b] → ℝ is continuous, then f is bounded. Let I := [a,b] be a closed bounded interval and f : A → ℝ be continuous on A, then f is bounded on I.

Bounded Function

If f : [a,b] → ℝ, f is bounded if its range f([a,b]) = {f(x) : a ≤ x ≤ b} is a bounded subset of ℝ. ∃M > 0 s.t. |f(x)| < M for ∀x ∈ [a,b].

Continuity of Rational Functions

If p and q are polynomial functions on ℝ, then there are at most a finite number α1, ..., αm of real roots of q. If x ∉ {α1, ..., αm} then q(x) ≠ 0 so that a rational function is continuous at every real number for which it is defined.

Continuity of Polynomial Functions

If p is a polynomial function, so that p(x) = anx^n + an-1x^(n-1) + ... + a1x + a0 for ∀x ∈ ℝ then it follows that p(c) = lim[x→c] p for ∀c ∈ ℝ. Thus a polynomial function is continuous on ℝ.

Caratheodory's Theorem

Let f be defined on an interval I containing the point c. Then f is differentiatiable at c if and only if 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. Thus φ(c) = f'(c).

Continuity of Trigonometric Functions

Sine and cosine are continuous on ℝ. Tangent, cotangent, secant, and cosecant are continuous where they are defined.

Weierstrass Approximation Theorem

Suppose A := [a,b] and f : A → ℝ is a continuous function. If ε > 0 is given, then there exists a polynomial function pε s.t. |f(x) - pε(x)| < ε for ∀x ∈ A.

Continuity of Absolute Value

Suppose A ⊆ ℝ, f : A → ℝ, and |f| be defined by |f|(x) := |f(x)| for x ∈ A. i) if f is continuous at point c ∈ A, then |f| is continuous at c. ii) If f is continuous on A, then |f| is continuous on A.

Continuity of Square Root

Suppose A ⊆ ℝ, f : A → ℝ, f(x) ≥ 0, and √f be defined by (√f)(x) := √(f(x)) for x ∈ A. i) if f is continuous at point c ∈ A, then √f is continuous at c. ii) If f is continuous on A, then √f is continuous on A.

Continuity of Algebraic Functions

Suppose A ⊆ ℝ, f, g : A → ℝ, and b ∈ ℝ. Let ∈ A and f and g are continuous at c. i) f + g, f - g, fg, and bf are continuous at c. ii) If h : A → ℝ is continuous at c ∈ A and if h(x) ≠ 0 for ∀x ∈ A, then the quotient f/h is continuous at c.

ε-δ Criterion of Integrable

Suppose I := [a, b] and f : I → ℝ be a bounded function on I. Then f is Darboux integrable on I if and only if for each ε > 0 there is a partition Pε of I such that U(f; Pε) - L(f; Pε) < ε.

Darboux Integrable

Suppose I = [a, b] and f : I → ℝ is a bounded function. Then f is Darboux Integrable on I if L(f) = U(f).

Chain Rule

Suppose I and J are intervals, f : J → I, g : I → ℝ, h : J → ℝ, h = g ₀ f, c ∈ J, f is differentiable at c and g is differentiable at f(c). Then h is differentiable at c and h'(c) = g'(f(c))f'(c).

ε-δ Criterion of Differentiable

Suppose I ⊆ ℝ is an interval, f : I → ℝ, and c ∈ I. L ∈ ℝ is a derivative of f at c if for ∀ε > 0, ∃δ(ε) > 0 s.t. if x ∈ I satisfies 0 < |x - c| < δ(ε), then |((f(x) - f(c))/(x - c)) - L| < ε.

ε-δ Criterion of Continuity

Suppose c ∈ A and f : A → ℝ. Then f is continuous at c if ∀ε > 0, ∃δ = δ(ε) such that if |x - c| < δ and x ∈ A, then |f(x) - f(c)| < ε.

Sequential Criterion for Continuity

Suppose c ∈ A and f : A → ℝ. Then the following conditions are equivalent: i) f is continuous at c ii) ∀xn ∈ A with xn → c it is true that f(xn) → f(c)

Limit Criterion of Continuity

Suppose c ∈ A is a cluster point of A, and f : A → ℝ. Then f is continuous at c if and only if lim[x→c] f(x) = f(c).

First Derivative Test

Suppose f : (a, b) → ℝ is continuous. Consider s ∈ (a, b). i) Suppose ∃δ > 0 s.t. if s - δ < x < s, then f'(x) ≤ 0 and if s < x < s + δ, then f'(x) ≥ 0. Then f has a relative minimum at s. ii) Suppose ∃δ > 0 s.t. (s - δ, s + δ) ⊂ (a, b) and if s - δ < x < s, then f'(x) ≥ 0 and if s < x < s + δ, then f'(x) ≤ 0. Then f has a relative maximum at s.

Continuous Extensions at End Points

Suppose f : (a, b) → ℝ is continuous. The following are equivalent: i) f is uniformly continuous on (a, b) ii) There is a continuous extension of f to [a, b].

Preservation of Intervals Theorem

Suppose f : A → ℝ is continuous on A, then the set f(A) is an interval.

Restriction of Functions

Suppose f : A → ℝ, A1 ⊂ A. Then restriction of f to A1 is g : A1 → ℝ given by g(x) = f(x), for ∀x ∈ A1 (written as g = f|A1.

Uniform Continuity

Suppose f : A → ℝ. If ∀ε > 0, ∃δ = δ(ε) > 0 s.t. if x ∈ A and u ∈ A and |x - u| < δ, then |f(x) - f(u)| < ε.

Nonuniform Continuity Criterion

Suppose f : A → ℝ. The following are equivalent: i) f is not uniformly continuous on A. ii) ∃x0 > 0 s.t. ∀δ > 0, ∃xδ ∈ A and uδ ∈ A with |xδ - uδ| < δ and |f(xδ) - f(uδ)| ≥ ε0. iii) ∃ε0 > 0 and sequences xn ∈ A and un ∈ A s.t. xn - un → 0 and |f(xn) - f(un)| ≥ ε0.

Extension of Functions

Suppose f : A1 → ℝ and A ⊂ A1. Then F : A → ℝ is an extension of f to A if F(x) = f(x), for ∀x ∈ A1 (written as ↔F|A1 = f.

Interior Extremum Theorem

Suppose f : I → ℝ, c is an interior point of I, f has a relative extremum at c. If f'(c) exists, then f'(c) = 0.

Sum Rule

Suppose f : I → ℝ, c ∈ I, f'(c) and g'(c) exists. Then (f+g)'(c) = f'(c) + g'(c).

Quotient Rule

Suppose f : I → ℝ, c ∈ I, f'(c) and g'(c) exists. Then (f/g)'(c) = ((g(c)f'(c) - f(c)g'(c))/(g(c)^2)).

Product Rule

Suppose f : I → ℝ, c ∈ I, f'(c) and g'(c) exists. Then (fg)'(c) = f(c)g'(c) + f'(c)g(c).

Limit Criterion of Differentiable

Suppose f : I → ℝ, c ∈ I. The derivative of f at c is f'(c) = lim[x→c]((f(x) - f(c))/(x-c)), where the limit exists.

Increasing / Decreasing

Suppose f : I → ℝ. f is {increasing, decreasing} on I if whenever x1 < x2 in I, when {f(x1) ≤ f(x2), f(x1) ≥ f(x2)}. It is strictly if they cannot be equivalent.

Chain Rule II

Suppose f : J → I and g : I → ℝ are both differentiable, and h : J → ℝ s.t. h = g ₀ f, then h'(x) = g'(f(x)f'(x) for ∀x ∈ J.

Additivity Theorem

Suppose f : [a, b] → ℝ is bounded and c ∈ (a, b). i) If f is integrable on [a, c] and on [c, b], then f is integrable on [a, b] and ∫[a, b] f(x)dx + ∫[c, b] f(x)dx. ii) If f is integrable on [a, b], then f is integrable on [a, c] and on [c, b].

Mean Value Theorem (MVT)

Suppose f : [a, b] → ℝ is continuous and differentiable on (a, b). Then ∃c ∈ (a, b) s.t. f'(c) = ((f(b) - f(a))/(b - a)).

Squeze Theorem of Integrability

Suppose f : [a, b] → ℝ. Then f ∈ R[a, b] if and only if for every ε > 0 there exists functions α(ε) and ω(ε) in R[a, b] with α(ε) ≤ f(x) ≤ ω(ε) ∀x ∈ [a, b], s.t. ∫[a,b](ω(ε) - α(ε) < ε.

Location of Roots Theorem

Suppose f : [a,b] → ℝ is continuous and f(a)f(b) < 0. Then ∃c ∈ [a,b] s.t. f(c) = 0.

Ranges of Continuous Functions on Closed Intervals

Suppose f : [a,b] → ℝ is continuous, m := min{f(x) : a ≤ x ≤ b}, M := max{f(x) : a ≤ x ≤ b}. Then f([a,b]) = [m,M].

Bolzano's Intermediate Value Theorem (IVT)

Suppose f : [a,b] → ℝ is continuous. Suppose k ∈ ℝ between f(a) and f(b), that is either f(a) < k < f(b) or f(a) > k > f(b). Then ∃c ∈ [a,b] s.t. f(c) = k.

The Maximum-Minimum Theorem

Suppose f : [a,b] → ℝ is continuous. Then f([a,b]) has a largest and smallest number. Specifically, ∃x' ∈ [a,b] and ∃x'' ∈ [a,b] s.t. f(x') ≤ f(x) ≤ f(x'') for ∀x ∈ [a,b].

Rolle's Theorem

Suppose f : |a, b] → ℝ is continuous and differentiable on (a, b), and f(a) = 0 = f(b), then ∃c ∈ (a, b) s.t. f'(c) = 0.

Riemann Integrable

Suppose f: [a, b] → ℝ, ∃L ∈ ℝ s.t. ∀ε >0, ∃δε > 0 s.t. if P⁰ is any tagged partition of [a, b] with ||P⁰|| < δε, then |S(f; P⁰) - L| < ε.

Power Rule

Suppose n ∈ ℕ, let f(x) = x^n. Then f'(c) = nc^(n-1), for ∀c ∈ ℝ.

Upper and Lower Integral

The upper integral of f is U(f) = inf{U(f, P) : P is a partition of [a, b]}. The lower integral of f is L(f) = sup{L(f, P) : P is a partition of [a, b]}. And L(f, P) ≤ U(f, P).

Continuous Function

f : A → ℝ is continuous (continuous on A) if f is continuous at each c ∈ A.

Lipschitz Condition

f : A → ℝ where ∃k > 0 s.t. |f(x) - f(u)| ≤ k|x - u|, for ∀u,x ∈ A.

Differentiable on an Interval

f : I → ℝ where f is differentiable at each point of I.

Upper and Lower Sum

∆xi = xi - xi-1, the length of Ii. Mesh (or norm) of P is ||P|| = max[i=1→n] ∆xi. Let Mi = sup{f(x) : x ∈ Ii} and mi = inf{f(x) : x ∈ Ii}. The upper sum for f associated with P is U(f, P) = ∑[i=1, n] Mi∆xi. The lower sum for f associated with P is L(f, P) = ∑[i=1, n] mi∆xi.