RAC integration definitions Flashcards
upper bound
X⊆ℝ. we call M∈ℝ an upper bound for X if x≤M for all x∈X
supremum
X⊆ℝ. we call N∈ℝ the supremum of X if:
(a) N is an upper bound
(b) N≤M for all upper bounds
ITS THE LOWEST UPPERBOUND
lower bound
X⊆ℝ. we call m∈ℝ a lower bound for X if x≥M for all x∈X
infimum
X⊆ℝ. we call N∈ℝ the infimum of X if:
(a) N is an lower bound
(b) N≥M for all lower bounds
ITS THE BIGGEST UPPER BOUND
bounded
a function f: X->ℝis called bounded if there exists M∈ℝ such that |f(x)|≤M for all x∈X
partition
A partition of [a,b] is a finite set p={x₀, x₁, x₂, x₃, … xₙ} such that a = x₀ < x₁ < x₂ < x₃ < … < xₙ = b
Lower Sum
set mᵢ = inf{ f(x) | xᵢ₋₁ - xᵢ}
L(f,p) = Σmᵢ(xᵢ-xᵢ₋₁)
Upper Sum
set Mᵢ = sup{ f(x) | xᵢ₋₁ - xᵢ}
U(f,p) = ΣMᵢ(xᵢ-xᵢ₋₁)
Lower Integral
∫ᵇₐf = sup {L(f,p)| p is a partition of [a,b]
—–
Upper Integral
___
∫ᵇₐf = inf {U(f,p)| p is a partition of [a,b]
integrable
we call f: [a,b]->ℝ integrable if lower integral = upper integral.
integral of a function
integral of f = lower integral of f = upper integral of f
ε-p definiton of integration
f: [a,b]->[0,∞) where f is bounded and non-negative is integrable if and only if:
For each ε>0, there exists a partition of P such that U(f,p)-L(f,p)
Continuity
f is continuous at y∈(a,b) if lim x->y f(x) = f(y)
Continuity ε-𝛿 definition
for each ε>0 there exists 𝛿>0 such that |x-y| |f(x)-f(y)|
integral of real (inc negative) functions
if f:[a,b]->ℝ is bounded, then define
f⁺(x) = {f(x) if f(x)≥0, 0 otherwise}
f⁻(x) = {0 otherwise, f(x) if f(x)<0}
so f = f⁺ + f⁻. we say that f is integrable on [a,b] if both f⁺ and -f⁻ are integrable and then define the integral to be:
∫ᵇₐf = ∫ᵇₐ (f⁺) - ∫ᵇₐ (-f⁻)
Axiom of completeness
If X is a non empty set of real numbers that has at least one upper boundm then the supremum of X exists. (Same for lower bound/infimum)
Monotonic
A function f:[a,b] -> ℝ is called monotonic on [a,b] if either x≤y => f(x) ≤ f(y) (increasing) or x≥y => f(x) ≥ f(y) (decreasing) for all x,y∈[a,b]
continuous at a point
A function f: [a,b]->ℝ is called continuous at a point c∈(a,b) if lim x->c f(x) = f(c)
differentiable at a point
A function f: [a,b]->ℝ is called differentiable at a point c∈(a,b) if the limit
f’(c) := lim h->0 f(c+h) - f(c) / h exists
differentiable
We say f is differentiable on (a,b) if its differentiable at every point c∈(a,b)
integrability of negative functions
f = f⁺ +f⁻, we say that f is integrable on [a,b] if both f⁺ and f⁻ are integrable and then define the integral to be:
∫ᵇₐf = ∫ᵇₐ (f⁺) - ∫ᵇₐ (-f⁻)
improper integral with upper bound infinity
if f: [a,∞)->ℝ is bounded and integrable on each sub interval [a,t] for all t>a, then
∫∞ₐf := limit as t->∞ ∫ᵗₐf
provided the limit exists
convergent improper integral
An improper integral is called convergent if the integral is finite
