Analysis II

Question 1

Define a limit point

Answer 1

A limit point of a set E ⊆ R is a real number p such that ∀δ > 0, ∃x ∈ E such that 0 < |x - p|< δ

Question 2

Define a closed set

Answer 2

E is closed if every limit point of E lies in E

Question 3

Define an isolated point

Answer 3

A point that is not a limit point

Question 4

Define an open set

Answer 4

E is open if the complement of E is closed

Question 5

Define convergence of a function

Answer 5

f(x) converges to L as n -> p if ∀ε > 0, ∃δ > 0 such that ∀x ∈ E, 0 <|x - p|< δ implies that |f(x) - L|< ε

Question 6

Define convergence of a function to infinity

Answer 6

f(x) converges to infinity as n -> p if ∀M ∈ R, ∃δ > 0 such that ∀x ∈ E, 0 <|x - p|< δ implies that f(x) > M.

Question 7

Define a right limit

Answer 7

If L is a right limit of f, ∀ε > 0, ∃δ > 0 such that ∀x ∈ E, p ≤ x < p + δ implies that |f(x) - L|< ε

Question 8

Define a left limit

Answer 8

If L is a left limit of f, ∀ε > 0, ∃δ > 0 such that ∀x ∈ E, p − δ < x ≤ p implies that |f(x) - L|< ε

Question 9

Define convergence at infinity

Answer 9

f(x) converges to L as n -> ∞ if ∀ε > 0, ∃N ∈ R such that ∀x ∈ E, x > N implies that |f(x) - L|< ε

Question 10

Define what it means for a function to be continuous at p

Answer 10

If f is continuous at p, ∀ε > 0, ∃δ > 0 such that ∀x ∈ E, |x - p| < δ implies that |f(x) - f(p)|< ε.

Question 11

Define a continuous function

Answer 11

If f: E -> R is continuous at all points p in E, f is continuous

Question 12

Define a right continuous function

Answer 12

If f is right continuous at p, ∀ε > 0, ∃δ > 0 such that ∀x ∈ E, p ≤ x < p + δ implies that |f(x) - f(p)|< ε.

Question 13

Define a left continuous function

Answer 13

If f is right continuous at p, ∀ε > 0, ∃δ > 0 such that ∀x ∈ E, p − δ < x ≤ p implies that |f(x) − f(p)| < ε).

Question 14

Define a bounded function

Answer 14

f is bounded if ∃M such that ∀x, |f(x)| ≤ M

Question 15

Define a uniformly continuous function

Answer 15

If f is uniformly continuous, ∀ε > 0, ∃δ > 0: ∀p ∈ E and ∀x ∈ E, |x - p| < δ implies that |f(x) - f(p)|< ε.

Question 16

Define a Lipschitz continuous function

Answer 16

If f: E -> R is Lipschitz continuous, ∃K: ∀x,y ∈ E |f(x) - f(y)|≤ K|x - y|.

Question 17

Define uniform convergence

Answer 17

f_n converges uniformly if ∀ε > 0, ∃N: ∀n ≥ N, ∀x ∈ E,|f_n(x) - f(x)|< ε.

Question 18

Define a function being differentiable at p

Answer 18

f is differentiable at p with derivative L if lim(x->p) of [f(x)-f(p)]/[x-p] = L

Question 19

Define a differentiable function

Answer 19

f: E -> R is differentiable if f is differentiable for all p ∈ E.

Question 20

Define a local maximum

Answer 20

p ∈ E is a local maximum of f if there exists δ > 0 such that f(x) ≤ f(p) for all x ∈ (p - δ, p + δ)∩E.

Question 20

Define a local minimum

Answer 20

p ∈ E is a local minimum of f if there exists δ > 0 such that f(x) ≥ f(p) for all x ∈ (p - δ, p + δ)∩E.

Question 21

Define an analytic function

Answer 21

f is analytic at p if ∃δ > 0 such that ∀ |h| < δ, f(p+h) can be written as a power series.