Real Math Partial 1

Definition – Neighborhood (Points in ℝ) (important)

A neighborhood of x₀ ∈ ℝ with radius ε > 0 is Bε(x₀) = {x ∈ ℝ : x₀ − ε < x < x₀ + ε}.

Definition – Neighborhood (Points in ℝⁿ) (important)

A neighborhood of x₀ ∈ ℝⁿ with radius ε > 0 is Bε(x₀) = {x ∈ ℝⁿ : ‖x − x₀‖ < ε}.

Theorem – Double complement (Sets) (important)

Let A ⊆ U. Then (Aᶜ)ᶜ = A.

Theorem – De Morgan’s laws (Sets) (important)

Let A,B ⊆ U. Then (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ and (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ.

Definition – Upper bound (Nonempty subsets of ℝ) (important)

Let A ⊆ ℝ, A ≠ ∅. A number h ∈ ℝ is an upper bound of A if ∀x ∈ A, x ≤ h. A* is the set of upper bounds of A.

Definition – Lower bound (Nonempty subsets of ℝ) (important)

Let A ⊆ ℝ, A ≠ ∅. A number k ∈ ℝ is a lower bound of A if ∀x ∈ A, k ≤ x. A₍*₎ is the set of lower bounds of A.

Definition – Supremum (Nonempty subsets of ℝ) (important)

Let A ⊆ ℝ, A ≠ ∅. The supremum of A is the least upper bound of A, so sup A = min A*.

Definition – Infimum (Nonempty subsets of ℝ) (important)

Let A ⊆ ℝ, A ≠ ∅. The infimum of A is the greatest lower bound of A, so inf A = max A₍*₎.

Definition – Maximum (Nonempty subsets of ℝ) (important)

Let A ⊆ ℝ, A ≠ ∅. A point x̂ ∈ ℝ is the maximum of A if x̂ ∈ A and ∀x ∈ A, x ≤ x̂.

Definition – Minimum (Nonempty subsets of ℝ) (important)

Let A ⊆ ℝ, A ≠ ∅. A point x̂ ∈ ℝ is the minimum of A if x̂ ∈ A and ∀x ∈ A, x̂ ≤ x.

Definition – Bounded above (Nonempty subsets of ℝ) (important)

Let A ⊆ ℝ, A ≠ ∅. A is bounded above if ∃h ∈ ℝ such that ∀x ∈ A, x ≤ h.

Definition – Bounded below (Nonempty subsets of ℝ) (important)

Let A ⊆ ℝ, A ≠ ∅. A is bounded below if ∃k ∈ ℝ such that ∀x ∈ A, k ≤ x.

Definition – Bounded set (Nonempty subsets of ℝ) (important)

Let A ⊆ ℝ, A ≠ ∅. A is bounded if it is bounded above and below; equivalently, ∃h,k ∈ ℝ such that ∀x ∈ A, k ≤ x ≤ h; equivalently, ∃M > 0 such that ∀x ∈ A, |x| < M.

Theorem – Completeness theorem (Nonempty subsets of ℝ) (important)

Let A ⊆ ℝ, A ≠ ∅. If A is bounded above, then sup A exists. If A is bounded below, then inf A exists.

Theorem – Characterization of the supremum using left neighborhoods (Nonempty subsets of ℝ) (important)

Let A ⊆ ℝ and a ∈ ℝ. Then a = sup A if and only if: (1) ∀x ∈ A, x ≤ a; and (2) ∀ε > 0, ∃x ∈ A such that x ∈ (a − ε,a].

Definition – Linear combination (Vectors in ℝⁿ) (important)

For α₁,…,αₘ ∈ ℝ and x₁,…,xₘ ∈ ℝⁿ, α₁x₁ + ⋯ + αₘxₘ ∈ ℝⁿ is the linear combination of x₁,…,xₘ with coefficients or weights α₁,…,αₘ.

Definition – Inner product or scalar product (Vectors in ℝⁿ) (important)

For x = (x₁,…,xₙ), y = (y₁,…,yₙ) ∈ ℝⁿ, x · y = x₁y₁ + ⋯ + xₙyₙ ∈ ℝ.

Definition – Norm (Vectors in ℝⁿ) (important)

For x = (x₁,…,xₙ) ∈ ℝⁿ, ‖x‖ = √(x₁² + ⋯ + xₙ²) = √(x · x).

Theorem – Pythagorean theorem (Vectors in ℝⁿ) (important)

If x,y ∈ ℝⁿ and x ⟂ y, then ‖x‖² + ‖y‖² = ‖x + y‖².

Definition – Euclidean distance (Points in ℝⁿ) (important)

For x = (x₁,…,xₙ), y = (y₁,…,yₙ) ∈ ℝⁿ, d(x,y) = ‖y − x‖ = √((y₁ − x₁)² + ⋯ + (yₙ − xₙ)²).

Definition – Segment (Points in ℝⁿ) (important)

The segment with endpoints x,y ∈ ℝⁿ is {z ∈ ℝⁿ : z = αx + βy, α,β ∈ [0,1], α + β = 1}, or equivalently {z ∈ ℝⁿ : z = αx + (1 − α)y, α ∈ [0,1]}.

Definition – Convex set or interval (Subsets of ℝ) (important)

Let A ⊆ ℝ, A ≠ ∅. A is an interval if ∀x,y ∈ A, {z ∈ ℝ : x ≤ z ≤ y} ⊆ A; equivalently, ∀x,y ∈ A, ∀α ∈ [0,1], αx + (1 − α)y ∈ A.

Definition – Convex set (Subsets of ℝⁿ) (important)

Let A ⊆ ℝⁿ, A ≠ ∅. A is convex if ∀x,y ∈ A, ∀α ∈ [0,1], αx + (1 − α)y ∈ A.

Definition – Interior point (Subsets of ℝⁿ) (important)

Let A ⊆ ℝⁿ and x₀ ∈ ℝⁿ. x₀ is an interior point of A if ∃B(x₀) such that B(x₀) ⊆ A.

Definition – Exterior point (Subsets of ℝⁿ) (important)

Let A ⊆ ℝⁿ and x₀ ∈ ℝⁿ. x₀ is an exterior point of A if ∃B(x₀) such that B(x₀) ⊆ Aᶜ.

Definition – Boundary point (Subsets of ℝⁿ) (important)

Let A ⊆ ℝⁿ and x₀ ∈ ℝⁿ. x₀ is a boundary point of A if ∀B(x₀), B(x₀) ∩ A ≠ ∅ and B(x₀) ∩ Aᶜ ≠ ∅.

Definition – Limit or accumulation point (Subsets of ℝⁿ) (important)

Let A ⊆ ℝⁿ and x₀ ∈ ℝⁿ. x₀ is a limit or accumulation point of A if ∀B(x₀), ∃x₁ ≠ x₀ such that x₁ ∈ B(x₀) ∩ A.

Definition – Isolated point (Subsets of ℝⁿ) (important)

Let A ⊆ ℝⁿ and x₀ ∈ ℝⁿ. x₀ is an isolated point of A if ∃B(x₀) such that B(x₀) ∩ A = {x₀}.

Definition – Open set (Subsets of ℝⁿ) (important)

Let A ⊆ ℝⁿ. A is open if every point of A is an interior point of A.

Definition – Closed set (Subsets of ℝⁿ) (important)

Let A ⊆ ℝⁿ. A is closed if every boundary point of A belongs to A.

Definition – Bounded set (Subsets of ℝⁿ) (important)

Let A ⊆ ℝⁿ, A ≠ ∅. A is bounded if ∃B(0) such that A ⊆ B(0).

Definition – Function (Mappings f : A → B) (important)

f : A → B is a function if ∀x ∈ A, ∃!y ∈ B such that y = f(x).

Definition – Surjective function (Mappings f : A → B) (important)

Let f : A → B. f is surjective if ∀y ∈ B, ∃x ∈ A such that y = f(x); equivalently, f(A) = B.

Definition – Injective function (Mappings f : A → B) (important)

Let f : A → B. f is injective if ∀x₁,x₂ ∈ A, x₁ ≠ x₂ ⇒ f(x₁) ≠ f(x₂); equivalently, f(x₁) = f(x₂) ⇒ x₁ = x₂ for all x₁,x₂ ∈ A.

Definition – Increasing function (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ. f is increasing if ∀x₁,x₂ ∈ A, x₁ < x₂ ⇒ f(x₁) ≤ f(x₂).

Definition – Strictly increasing function (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ. f is strictly increasing if ∀x₁,x₂ ∈ A, x₁ < x₂ ⇒ f(x₁) < f(x₂).

Theorem – Strict monotonicity and order isomorphism (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ. f is strictly increasing on A if and only if ∀x₁,x₂ ∈ A, x₁ ≤ x₂ ⇔ f(x₁) ≤ f(x₂).

Definition – Strictly convex function (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ, where A is convex. f is strictly convex if ∀α ∈ (0,1), ∀x₁,x₂ ∈ A with x₁ ≠ x₂, f(αx₁ + (1 − α)x₂) < αf(x₁) + (1 − α)f(x₂).

Definition – Convex function (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ, where A is convex. f is convex if ∀α ∈ [0,1], ∀x₁,x₂ ∈ A, f(αx₁ + (1 − α)x₂) ≤ αf(x₁) + (1 − α)f(x₂).

Definition – Strictly quasi-convex function (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ, where A is convex. f is strictly quasi-convex if ∀α ∈ (0,1), ∀x₁,x₂ ∈ A with x₁ ≠ x₂, f(αx₁ + (1 − α)x₂) < max{f(x₁),f(x₂)}.

Definition – Quasi-convex function (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ, where A is convex. f is quasi-convex if ∀α ∈ [0,1], ∀x₁,x₂ ∈ A, f(αx₁ + (1 − α)x₂) ≤ max{f(x₁),f(x₂)}.

Definition – Strictly quasi-concave function (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ, where A is convex. f is strictly quasi-concave if ∀α ∈ (0,1), ∀x₁,x₂ ∈ A with x₁ ≠ x₂, f(αx₁ + (1 − α)x₂) > min{f(x₁),f(x₂)}.

Definition – Quasi-concave function (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ, where A is convex. f is quasi-concave if ∀α ∈ [0,1], ∀x₁,x₂ ∈ A, f(αx₁ + (1 − α)x₂) ≥ min{f(x₁),f(x₂)}.

Theorem – Injectivity and strict monotonicity (Monotonic functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ be monotonic on A. Then f is injective or invertible on A if and only if f is strictly monotonic.

Definition – Global maximizer (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ and x₀ ∈ A. x₀ is a global maximizer if ∀x ∈ A, f(x) ≤ f(x₀). max₍x∈A₎ f(x) represents the maximum, while arg max₍x∈A₎ f(x) represents the maximizers.

Definition – Strong global maximizer (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ and x₀ ∈ A. x₀ is a strong or strict global maximizer if ∀x ∈ A with x ≠ x₀, f(x) < f(x₀).

Definition – Local maximizer (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ and x₀ ∈ A. x₀ is a local maximizer if ∃B(x₀) such that ∀x ∈ B(x₀) ∩ A, f(x) ≤ f(x₀).

Definition – Bounded function (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ. f is bounded if its range f(A) is bounded: ∃h ∈ ℝ such that ∀x ∈ A, f(x) ≤ h, and ∃k ∈ ℝ such that ∀x ∈ A, f(x) ≥ k; equivalently, ∃M > 0 such that ∀x ∈ A, |f(x)| < M.

Theorem – Fenchel’s local-global extremum property (Concave and convex functions) (important)

Let f : A ⊆ ℝ → ℝ, where A is convex. If f is concave on A, every local maximizer is a global maximizer. If f is convex on A, every local minimizer is a global minimizer.

Definition – Composite function (Mappings f : A → B and g : C → D) (important)

Let f : A → B and g : C → D. If f(A) ⊆ C, then g ∘ f : A → D is defined by (g ∘ f)(x) = g(f(x)) for every x ∈ A.

Theorem – Preservation of maximizers and minimizers under strictly increasing compositions (Functions) (important)

Let f : A ⊆ ℝ → ℝ and let g : B ⊆ ℝ → ℝ be strictly increasing, with f(A) ⊆ B. Then arg max₍x∈A₎ f(x) and arg max₍x∈A₎ g(f(x)) have the same solutions; likewise, arg min₍x∈A₎ f(x) and arg min₍x∈A₎ g(f(x)) have the same solutions.

Definition – Inverse function (Mappings f : A → B) (important)

Let f : A → B. f is invertible on A if ∃f⁻¹ : f(A) → A such that f(f⁻¹(x)) = x for every x ∈ f(A) and f⁻¹(f(x)) = x for every x ∈ A.

Definition – Level set (Functions f : A ⊆ ℝⁿ → ℝ) (important)

Let f : A ⊆ ℝⁿ → ℝ. A level set of height or level k is {x ∈ A : f(x) = k}.

Definition – Upper contour set (Functions f : C ⊆ ℝⁿ → ℝ) (important)

Let f : C ⊆ ℝⁿ → ℝ, where C is convex. The upper contour set of level k is Uₖ = {x ∈ C : f(x) ≥ k}.

Theorem – Convexity of contour sets of concave and convex functions (Functions f : C ⊆ ℝⁿ → ℝ) (important)

Let C be convex and let f : C ⊆ ℝⁿ → ℝ. If f is concave, all its upper contour sets {f ≥ k} are convex. If f is convex, all its lower contour sets {f ≤ k} are convex.

Theorem – Characterization of quasi-concavity and quasi-convexity using contour sets (Functions f : C ⊆ ℝⁿ → ℝ) (important)

Let f : C ⊆ ℝⁿ → ℝ. f is quasi-concave if and only if all its upper contour sets are convex. f is quasi-convex if and only if all its lower contour sets are convex.

Theorem – Convexity of contour sets of monotonic functions (Functions on intervals in ℝ) (important)

Let f : A → ℝ be monotonic, where A is an interval. Then all its upper and lower contour sets are convex.

Theorem – Increasing compositions preserve concavity and quasi-concavity (Functions) (important)

Let g : C ⊆ ℝ → ℝ and f : D ⊆ ℝ → ℝ, where C and D are convex and g(C) ⊆ D. If g is concave and f is concave and increasing, then f ∘ g : C → ℝ is concave. If g is quasi-concave and f is increasing, then f ∘ g : C → ℝ is quasi-concave.

Definition – Strictly increasing sequence (Sequences of real numbers) (important)

A sequence (aₙ) is strictly increasing if ∀n ∈ ℕ, aₙ < aₙ₊₁; equivalently, ∀n ≥ 1, aₙ₋₁ < aₙ.

Definition – Convergent sequence (Sequences of real numbers) (important)

A sequence (aₙ) converges to L ∈ ℝ if limₙ→∞ aₙ = L: ∀B(L), ∃n̄ ∈ ℕ such that ∀n ≥ n̄, aₙ ∈ B(L); equivalently, ∀ε > 0, ∃n̄ ∈ ℕ such that ∀n ≥ n̄, L − ε < aₙ < L + ε; equivalently, ∀ε > 0, ∃n̄ ∈ ℕ such that ∀n ≥ n̄, |aₙ − L| < ε.

Definition – Convergence from above (Sequences of real numbers) (important)

A sequence (aₙ) converges to L from above if limₙ→∞ aₙ = L⁺: ∀B⁺(L), ∃n̄ ∈ ℕ such that ∀n ≥ n̄, aₙ ∈ B⁺(L); equivalently, ∀ε > 0, ∃n̄ ∈ ℕ such that ∀n ≥ n̄, L ≤ aₙ < L + ε.

Definition – Divergence to positive infinity (Sequences of real numbers) (important)

A sequence (aₙ) diverges to +∞ if limₙ→∞ aₙ = +∞: ∀B(+∞), ∃n̄ ∈ ℕ such that ∀n ≥ n̄, aₙ ∈ B(+∞); equivalently, ∀K > 0, ∃n̄ ∈ ℕ such that ∀n ≥ n̄, aₙ > K.

Definition – Irregular or oscillating sequence (Sequences of real numbers) (important)

If a sequence (aₙ) does not converge to a finite limit or diverge to +∞ or −∞, then limₙ→∞ aₙ does not exist and (aₙ) is irregular or oscillating.

Theorem – Comparison criterion or squeeze theorem (Sequences of real numbers) (important)

If aₙ ≤ bₙ ≤ cₙ eventually and limₙ→∞ aₙ = limₙ→∞ cₙ = L, then limₙ→∞ bₙ = L.

Theorem – Regularity of monotonic sequences (Sequences of real numbers) (important)

Every monotonic sequence (aₙ) is regular. Specifically: if (aₙ) is increasing and bounded, it converges to a finite limit L⁻; if it is increasing and unbounded, it diverges to +∞; if it is decreasing and bounded, it converges to a finite limit L⁺; if it is decreasing and unbounded, it diverges to −∞.

Theorem – Convergence characterized by distance (Sequences of real numbers) (important)

A sequence (aₙ) converges to a finite limit L if and only if d(aₙ,L) → 0⁺.

Theorem – Relationship between infinities and infinitesimals (Eventually positive sequences) (important)

Suppose (aₙ) is eventually positive. Then aₙ → +∞ if and only if 1/aₙ → 0⁺.

Theorem – Permanence of the sign (Sequences of real numbers) (important)

If aₙ ≥ 0 eventually and limₙ→∞ aₙ exists, then limₙ→∞ aₙ ≥ 0.

Theorem – Operations with limits (Regular sequences) (important)

Suppose (aₙ) and (bₙ) are regular, limₙ→∞ aₙ = L and limₙ→∞ bₙ = M, where L,M ∈ ℝ̄. If no indeterminate form occurs, then limₙ→∞(aₙ + bₙ) = L + M, limₙ→∞(aₙbₙ) = LM, and limₙ→∞(aₙ/bₙ) = L/M.

Definition – Little-o (Sequences of real numbers) (important)

aₙ = o(bₙ), meaning that aₙ is little-o of bₙ, if limₙ→∞(aₙ/bₙ) = 0.

Definition – Asymptotic equivalence (Sequences of real numbers) (important)

aₙ ∼ bₙ, meaning that aₙ is asymptotic to bₙ, if limₙ→∞(aₙ/bₙ) = 1.

Theorem – Characterization of asymptotic equivalence using little-o (Sequences of real numbers) (important)

aₙ ∼ bₙ if and only if aₙ = bₙ + o(bₙ).

Definition – Convergence of a vector sequence (Sequences in ℝⁿ) (important)

A sequence (xₙ) in ℝⁿ converges to L if ∀B(L), ∃n̄ ∈ ℕ such that ∀n ≥ n̄, xₙ ∈ B(L).

Theorem – Ratio criterion (Sequences of real numbers) (important)

Consider (aₙ), with aₙ ≠ 0 eventually. If limₙ→∞ |aₙ₊₁/aₙ| = q, where q ∈ ℝ and q < 1, then limₙ→∞ aₙ = 0.

Definition – Series and sequence of partial sums (Series of real numbers) (important)

Consider a sequence aₙ = a₀,a₁,a₂,…. The sequence of partial sums (Sₙ) is defined by S₀ = a₀, S₁ = a₀ + a₁, S₂ = a₀ + a₁ + a₂, …, Sₙ = a₀ + a₁ + ⋯ + aₙ. The series with general term aₙ is ∑ₙ₌₀^∞ aₙ = limₙ→∞ Sₙ, if this limit exists.

Definition – Behaviour of a series (Series of real numbers) (important)

If limₙ→∞ Sₙ = S, then ∑ₙ₌₀^∞ aₙ = S and the series converges to S. If limₙ→∞ Sₙ = +∞, then ∑ₙ₌₀^∞ aₙ = +∞ and the series diverges to +∞. If limₙ→∞ Sₙ does not exist, then ∑ₙ₌₀^∞ aₙ is irregular.

Theorem – Necessary condition for convergence (Series of real numbers) (important)

If ∑ₙ₌₀^∞ aₙ converges, then limₙ→∞ aₙ = 0.

Theorem – Regularity theorem (Series with eventually nonnegative terms) (important)

If aₙ ≥ 0 eventually, then ∑ₙ₌₀^∞ aₙ is regular and either converges to a finite value S or diverges to +∞.

Theorem – Asymptotic comparison criterion (Series with nonnegative terms) (important)

If aₙ,bₙ ≥ 0 eventually and aₙ ∼ bₙ, then ∑ₙ₌₀^∞ aₙ and ∑ₙ₌₀^∞ bₙ have the same behaviour: they both converge or both diverge to +∞.

Theorem – Comparison criterion (Series with nonnegative terms) (important)

If aₙ,bₙ ≥ 0 eventually and aₙ ≤ bₙ eventually, then: if ∑ₙ₌₀^∞ bₙ converges, ∑ₙ₌₀^∞ aₙ converges; if ∑ₙ₌₀^∞ aₙ diverges to +∞, ∑ₙ₌₀^∞ bₙ diverges to +∞.

Theorem – Absolute convergence test (Series of real numbers) (important)

If ∑ₙ₌₀^∞ |aₙ| converges, then ∑ₙ₌₀^∞ aₙ converges absolutely.

Definition – Continuity at a point (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ and x₀ ∈ A. If x₀ is a limit point of A, then f is continuous at x₀ if limₓ→x₀ f(x) = L and f(x₀) = L. If x₀ is an isolated point of A, then f is continuous at x₀.

Definition – Finite limit at a finite point (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ, where c is a limit point of A. limₓ→c f(x) = L if ∀U(L), ∃B(c) such that ∀x ∈ B(c) \ {c}, f(x) ∈ U(L).

Definition – Finite limit at positive infinity (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ, where A is unbounded above. limₓ→+∞ f(x) = L if ∀U(L), ∃B(+∞) such that ∀x ∈ B(+∞), f(x) ∈ U(L).

Theorem – Sequential characterization of limits (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ and let c ∈ ℝ be a limit point of A. Then limₓ→c f(x) = L ∈ ℝ̄ if and only if, for every sequence (xₙ) in A with xₙ ≠ c for every n and xₙ → c, we have f(xₙ) → L.

Theorem – Uniqueness of the limit (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ, where c ∈ ℝ is a limit point of A. If limₓ→c f(x) exists, then it is unique.

Theorem – Permanence of the sign (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ be defined on B(c) \ {c}. If f(x) ≥ 0 on B(c) \ {c} and limₓ→c f(x) exists, then limₓ→c f(x) ≥ 0.

Theorem – Operations with limits (Functions) (important)

Consider f(x) and g(x), with limₓ→c f(x) = L and limₓ→c g(x) = M, where L,M ∈ ℝ̄. If no indeterminate form occurs, then limₓ→c[f(x) + g(x)] = L + M, limₓ→c[f(x)g(x)] = LM, and limₓ→c[f(x)/g(x)] = L/M.

Theorem – Comparison criterion (Limits of functions) (important)

Consider f(x),g(x),h(x), defined on B(c) \ {c}. If f(x) ≤ g(x) ≤ h(x) on B(c) \ {c} and limₓ→c f(x) = limₓ→c h(x) = L, then limₓ→c g(x) = L.

Theorem – Direct characterization of continuity (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ and x₀ ∈ A. f is continuous at x₀ if and only if ∀U(f(x₀)), ∃B(x₀) such that ∀x ∈ B(x₀), f(x) ∈ U(f(x₀)); equivalently, ∀ε > 0, ∃δ = δ(ε) > 0 such that x₀ − δ < x < x₀ + δ implies f(x₀) − ε < f(x) < f(x₀) + ε.

Theorem – Sequential characterization of continuity (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ and x₀ ∈ A. f is continuous at x₀ if and only if, for every sequence (xₙ) of points of A, xₙ → x₀ implies f(xₙ) → f(x₀).

Theorem – Continuity and operations (Continuous functions) (important)

Let f,g : A ⊆ ℝ → ℝ be continuous. Then f + g is continuous, fg is continuous, and f/g is continuous wherever g(x) ≠ 0. If f : A ⊆ ℝ → ℝ and g : B ⊆ ℝ → ℝ are continuous and f(A) ⊆ B, then g ∘ f is continuous.

Theorem – Invertibility and strict monotonicity (Continuous functions on intervals) (important)

Let f : A ⊆ ℝ → ℝ, where A is an interval and f is continuous on A. Then f is invertible on A if and only if f is strictly monotonic on A.

Definition – Little-o (Functions) (important)

f(x) = o(g(x)) as x → c if limₓ→c[f(x)/g(x)] = 0.

Definition – Asymptotic equivalence (Functions) (important)

f(x) ∼ g(x) as x → c if limₓ→c[f(x)/g(x)] = 1.

Definition – Vertical asymptote (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ. The line x = c is a vertical asymptote if at least one of the following holds: limₓ→c⁻ f(x) = +∞ or −∞; limₓ→c⁺ f(x) = +∞ or −∞.

Definition – Horizontal asymptote (Functions f : A ⊆ ℝ → ℝ) (important)

Let f : A ⊆ ℝ → ℝ. The line y = L is a horizontal asymptote for f as x → +∞ if limₓ→+∞ f(x) = L, and it is a horizontal asymptote for f as x → −∞ if limₓ→−∞ f(x) = L.

Theorem – Zero-value or Bolzano’s theorem (Continuous functions on closed bounded intervals) (important)

Let f : A ⊆ ℝ → ℝ. If A = [a,b] is a closed and bounded interval, f is continuous on A, and f(a)f(b) ≤ 0, then ∃c ∈ [a,b] such that f(c) = 0.

Theorem – Existence and uniqueness of market equilibrium (Continuous demand and supply functions)

Consider a demand function D = D(p) : [a,b] → ℝ and a supply function S = S(p) : [a,b] → ℝ. Suppose D and S are continuous, D(a) ≥ S(a), and D(b) ≤ S(b). Then there exists a market equilibrium E = (p̄,q̄), where p̄ ∈ [a,b]. If D is strictly decreasing and S is strictly increasing, E is unique.

Theorem – Weierstrass’s theorem (Continuous functions on closed bounded sets) (important)

Let f : A ⊆ ℝ → ℝ. If A is closed and bounded and f is continuous on A, then ∃max₍x∈A₎ f(x) = M and ∃min₍x∈A₎ f(x) = m.