Analysis II // Flashcards
Triangle inequality
|a+b| ≤ |a| + |b|
|a-b| ≥ ||a| - |b||
Continuity - ε-δ
A function f: E → ℝ is continuous at c ∈ E if for any ε > 0 there exists a δ > 0 such that if x ∈ E and |x - c| < δ, then |f(x) - f(c)| < ε
δ-neighbourhood of c in E
x ∈ E with |x - c| < δ, i.e. E ∩ (c - δ, c + δ)
Sequential continuity
A function f: E → ℝ is sequentially continuous at c ∈ E if whenever (x_n) ∈ E and x_n → c as n → ∞, then f(x_n) → f(c) as n → ∞
Interval
A non-empty subset I of ℝ is an interval if whenever x,y ∈ I and x < y then [x,y] ∈ I
Open subset
A subset of A of ℝ is open if for every x ∈ A there exists ε > 0 such that (x - ε, x + ε) ⊂ A
Closed
A subset A of ℝ is closed if ℝ \ A is open
Injective
f(x) = f(y) ⇒ x = y
Surjective
For every a ∈ B there exists x ∈ A such that f(x) = a
Increasing
Let E ⊂ ℝ. A function f: E → ℝ is said to be increasing on E if x ≥ y ⇒ f(x) ≥ f(y)
Strictly increasing
Let E ⊂ ℝ. A function f: E → ℝ is said to be strictly increasing on E if x > y ⇒ f(x) > f(y)
Decreasing
Let E ⊂ ℝ. A function f: E → ℝ is said to be decreasing on E if x ≥ y ⇒ f(x) ≤ f(y)
Strictly decreasing
Let E ⊂ ℝ. A function f: E → ℝ is said to be strictly decreasing on E if x > y ⇒ f(x) < f(y)
Continuous limit
If f: (a,b) \ {c} → ℝ then we say that f(x) tends to α as x tends to c, if for every ε > 0 there exists a δ > 0 such that x ∈ (a,b) with 0 < |x-c| < δ ⇒ |f(x) - α| < ε
One-sided limit from the left
If f: (a,b) \ {c} → ℝ then we say that f(x) tends to α as x tends to c from below, if for every ε > 0 there exists a δ > 0 such that |f(x) - α| < ε for all x with c - δ < x < c
One-sided limit from the right
If f: (a,b) \ {c} → ℝ then we say that f(x) tends to α as x tends to c from above, if for every ε > 0 there exists a δ > 0 such that |f(x) - α| < ε for all x with c < x < c + δ
Limit as x → ∞
f(x) → α as x → ∞ if for every ε > 0 there exists an R such that |f(x) - α| < ε for all x > R
Limit to ∞
f(x) → ∞ as x → c if for every R there exists a δ > 0 such that f(x) > R for all x with 0 < |x - c| < δ
Differentiable
f: (a,b) → ℝ is differentiable at x₀ ∈ (a,b) if the limit
lim(x → x₀) (f(x) - f(x₀))/(x - x₀)
exists and is finite, denoted f’(x₀)
Continuously differentiable (C^1 on I)
f is continuously differentiable on an interval I if f is differentiable on I and f’ is continuous on I
Twice differentiable
f: (a,b) → ℝ is twice differentiable at x₀ ∈ (a,b) if f is differentiable on a neighbourhood of x₀, and f’ is differentiable at x₀ We then set f’‘(x₀) = (f’)’(x₀) the second derivative of f at x₀
n times differentiable
f: (a,b) → ℝ is n times differentiable at x₀ ∈ (a,b) if f is n - 1 times differentiable on a neighbourhood of x₀, and f^(n-1) is differentiable at x₀. We then set f^(n)(x₀) = (f^(n-1))’(x₀), the nth derivative of f at x₀
n times continuously differentiable (C^n on I)
f is n times continuously differentiable on an interval I if fⁿ exists and is continuous on I.
Smooth (C^∞ on I)
if f is n times continuously differentiable on I for every n ∈ ℕ we say that f is smooth (f ∈ C^∞(I))
Local maximum
Let f: (a,b) → ℝ and consider x₀ ∈ [a,b]. We say that f has a local maximum at x₀ if there exists a δ > 0 such that f(x) ≤ f(x₀) for all x ∈ [a,b] with |x - x₀| < δ
Local minimum
Let f: (a,b) → ℝ and consider x₀ ∈ [a,b]. We say that f has a local minimum at x₀ if there exists a δ > 0 such that f(x) ≥ f(x₀) for all x ∈ [a,b] with |x - x₀| < δ
Strict local maximum
Let f: (a,b) → ℝ and consider x₀ ∈ [a,b]. We say that f has a strict local maximum at x₀ if there exists a δ > 0 such that f(x) < f(x₀) for all x ∈ [a,b]{x₀} with |x - x₀| < δ
Strict local minimum
Let f: (a,b) → ℝ and consider x_0 ∈ [a,b]. We say that f has a strict local minimum at x_0 if there exists a δ > 0 such that f(x) > f(x_0) for all x ∈ [a,b]{x_0} with |x - x_0| < δ
Extremum
A maximum of minima
Taylor expansion/series of f about a
When f ∈ C^∞ the Taylor expansion/series of f about a is given by Σ(k=0, ∞) f^(k)(a)*(x - a)^k/k!
limsup
If (c_n) is a sequence, limsup (n → ∞) (c_n) = lim (n → ∞) (sup (j ≥ n) c_j)
liminf
If (c_n) is a sequence, liminf (n → ∞) (c_n) = lim (n → ∞) (inf (j ≥ n) c_j)
exp(x)
exp(x) = Σ (n=0, ∞) xⁿ /n!
Algebra of cts functions
If f,g: E → ℝ are both cts at c ∈ E then
i) f+g is cts at c
ii) fg is cts at c
Composition of cts functions
Suppose f: E ∈ ℝ is cts at c, f(E) ⊂ D, and g: D → ℝ is cts at f(c), then g ◦ f: E → ℝ is cts at c
Discontinuous function
A function f: E → ℝ is discontinuous at c ∈ E if there exists an ε > 0 such that for every δ > 0 there exists an x ∈ E such that
|x - c| < δ but |f(x) - f(c)| ≥ ε
Intermediate value theorem
Suppose that f is cts on [a,b] and that f(a) < f(b). Then for any g with f(a) < g < f(b) there exists c ∈ (a,b) such that f(c) = g
Fixed point theorem
Any cts function f: [a,b] → [a,b] has a fixed point, ie there exists an x* ∈ [a,b] such that f(x) = x
Extreme value theorem
If f: [a,b] → ℝ is cts then f is bounded above and below on [a,b], and attains its bounds: there exist x, x_ ∈ [a,b] such that
f(x_) = inf(x ∈ [a,b]) f(x)
and
f(x) = sup(x ∈ [a,b]) f(x)
Inverse function theorem
Let I be an interval and suppose that f: I → ℝ is cts and strictly monotonic. Then J = f(I) is an interval and f⁻¹: J → I is cts and strictly monotonic (in the same sense as f)
Sandwich rule for cts limits
If f(x) ≤ g(x) ≤ h(x) and lim(x → c) f(x) = lim(x → c) h(x) = α then lim(x → c) g(x) = α
Algebra of derivative
Suppose f,g: (a,b) → ℝ are diffble at x₀ ∈ (a,b). then
i) f+g is diffble at x₀ and
(f+g)’(x₀) = f’(x₀) + g’(x₀);
ii) fg is diffble at x₀ and
(fg)’(x₀) = f’(x₀)g(x₀) + f(x₀)g’(x₀)
iii) if g’(x₀) ≠ 0 then 1/g is diffble at x₀, and
(1/g)’(x₀) = -g’(x₀)/g²(x₀)
Carathéodory formation of differentiability
Let f: (a,b) → ℝ and take x₀ ∈ (a,b). Then d is diffble at x₀ with derivative f’(x₀) if and only if there exists a function φ that is cts at x₀ with φ(x₀) = f’(x₀) and
f(x) = f(x₀) + φ(x)(x - x₀)
Chain rule
Suppose that f: (a,b) → (c,d) is diffble at x₀ ∈ (a,b) and that g: (c,d) → ℝ is diffble at y₀ = f(x₀). Then g ◦ f: (a,b) → ℝ is diffble at x₀ and
(g ◦ f)’(x₀) = g’(f(x₀))f’(x₀)
Derivatives of inverses
Let f: [a,b] → [c,d] be a cts bijection. Suppose that f is diffble at x₀ ∈ (a,b) and f’(x₀) ≠ 0, then f⁻¹ is diffble at y₀ = f(x₀) and
(f⁻¹)’(y₀) = 1/f’(x₀)
L’Hôpital’s rule, pt 1
Let f,g: (a,b) → ℝ, and take x₀ ∈ (a,b). Suppose that lim(x → x₀) f(x) = lim(x → x₀) g(x) = 0, and that f and g are diffble at x₀ with g’(x₀) ≠ 0. Then
lim(x → x₀) f(x)/g(x) = f’(x₀)/g’(x₀)
Rolle’s theorem
Let f: [a,b] → ℝ be cts on [a,b], diffble on (a,b), with f(a) = f(b). Then there exists a point c ∈ (a,b) such that f’(c) = 0
Mean value theorem
Let f: [a,b] → ℝ be cts on [a,b] and diffble on (a,b). Then there exists a point c ∈ (a,b) such that
f’(c) = (f(b) - f(a))/(b - a)
Cauchy’s mean value theorem
If f,g are cts on [a,b] and diffble (a,b) then there exists a c ∈ (a,b) such that
(f(b) - f(a))g’(c) = (g(b) - g(a))f’(c)
L’Hôpital’s rule, better (one sided)
Suppose that f,g: (a,b) → ℝ are diffble at every point of (a,b) with g’(x) ≠ 0 for all x ∈ (a,b). Suppose that either
i) lim(x → b⁻) f(x) = lim(x → b⁻) g(x) = 0; or
ii) lim(x → b⁻) f(x) = ±∞ and lim(x → b⁻) g(x) = ±∞
then if lim(x → b⁻) f’(x)/g’(x) = ℓ
it follows that lim(x → b⁻) f(x)/g(x) = ℓ
where ℓ can be a real number or ±∞
