Real analysis

Question 1

what does it mean if A is bounded above

Answer 1

A set A ⊂ R is said to be bounded above, if A ⊂ (−∞, M], any such M would be a upper bound for A

Question 2

what is a neighbourhood of x

Answer 2

If x ∈ R, then a neighbourhood of x is any open interval (a, b) such that x ∈ (a, b)

Question 3

state the completeness axiom

Answer 3

• Every non-empty subset of ℝ which is bounded above has a unique supremum
• Every non-empty subset of ℝ which is bounded below has a unique infimum

Question 4

what does it mean for a set A to be countable

Answer 4

• It can either be finite OR a 1 to 1 bijective function from N to A

Question 5

what is required for a sequence to be a Cauchy sequence

Answer 5

∀ε > 0 ∃N ∈ N : ∀n, m ⩾ N : |an − am| < ε.

Question 6

are all Cauchy sequences convergent or are all convergent sequences Cauchy?

Answer 6

all convergent sequences are Cauchy, (Cauchy’s criterion)

Question 7

Bolzano-Weierstrass theorem
then in english.

Answer 7

Let {xₙ}∞ be a sequence of real numbers
such that xₙ ∈ [a, b] for all n ∈ N. Then {xₙ}∞
n=1 has a limit point in [a, b].
all bounded sequences have a convergent subsequence

Question 8

define an open set

Answer 8

• An interval (a,b) where every element in this interval has a neighbourhood
• every element is an interior point

Question 9

define a closed set

Answer 9

• a set where its complement is open

Question 10

define in mathematical symbols, what it means for the limit of f(x) as x → x₀

Answer 10

∀ε > 0 ∃δ > 0 : |x − x0| < δ ⇒ |f(x) − f(x0)| < ε

Question 11

in english?

Answer 11

For any neighbourhood U of f(x₀), there exists a neighbourhood V of x₀ such that if x ∈ V, then f(x) ∈ U

Question 12

how do we prove if a function is continuous about a range

Answer 12

1) Write out | f(x) - f(x₀) | < ε
2) Then choose a δ for which it works

Question 13

now give me the definition for the right limit

Answer 13

∀ε > 0 ∃δ > 0 : x0 < x < x0 + δ ⇒ |f(x) − y+| < ε.

Question 14

what is required for a point x₀ to be a point of removable discontinuity

Answer 14

if both right and left limits are finite and the same but f(x₀) is not defined or just not equal to these limits

Question 15

x₀ is a jump discontinuity if …

Answer 15

if both limits exist and are finite, but are not equal to each other.

Question 16

Infinite discontinuity

Answer 16

the point x0 is an infinite discontinuity of f, if at least one of the limits is infinite

Question 17

oscillatory dicontinuity

Answer 17

The point x0 is an oscillatory discontinuity of f, if at least one of the limits does not exist.

Question 18

give me a definition to do with convergence and continuity

Answer 18

lim as x tends to x0 of f(x) = y0 if and only if for all sequences xn that converges to x0, we have then that f(xn) tends to y0 as x tends to x0

Question 19

what does it mean for f to be continuous at x0 in terms of convergent sequences

Answer 19

A function f is continuous at x0 if and only if for every sequence of points xn ∈ (a, b) such that limn→∞ xn = x0 we have limn→∞ f(xn) = f(x0).

Question 19

what characteristic function of the set A

Answer 19

f(x): 1 when x ∈ A, 0 when x ∉ A

Question 20

what is the natural domain

Answer 20

the set of values x ∈ R such that the formula makes sense; that is, there is no division by zero, all square
roots and logarithm are taken of non-negative numbers, etc.

Question 21

state the boundedness theorem

Answer 21

• If f is a continuous function on a closed bounded interval, then f is bounded

Question 22

true or false: a function with a limit as x tends to infinity, that is continuous on an interval [a, infinity) is also bounded on that same interval

Answer 22

Let a ∈ R and let f be a continuous function on [a, ∞) such that the limit limx→∞ f(x) exists. Then f is bounded on [a, ∞).

Question 23

Maximum/Minimum Theorem

Answer 23

if a function is continuous on a closed bounded interval then its maximum and minimum is also on that same closed bounded interval

Question 24

intermediate value theorem

Answer 24

Let f be a continuous function on a closed bounded interval [a, b]. Then f attains every value between f(a) and f(b).

Question 25

fermat theorem

Answer 25

Let f be a differentiable function on the interval (a, b). If f has a local maximum or a local minimum at c ∈ (a, b) then f′(c) = 0.

Question 26

mean value theorem

Answer 26

Let f be a continuous function on [a, b]. If f is differentiable on (a, b) then there exists a point c in (a, b) at which f'(c) =f(b)−f(a)/b−a

Question 27

rolles theorem

Answer 27

Let f be a continuous function on [a, b] such that f(a) = f(b). If f is differentiable on (a, b) then there exists a point c in (a, b) at which f′(c) = 0.

Question 28

Let a ∈ R and let f be a continuous function on [a, ∞) such that the limit limx→∞ f(x) exists. Then

Answer 28

the function f is bounded between a and inf

Question 29

what does it mean for a function to be uniformly continuous on an interval Δ

Answer 29

- It means that for any ε > 0 there exists a δε > 0 such that ∀x, x′ ∈ ∆ : |x − x′| < δε ⇒ |f (x) − f (x′)| < ε. - clearly we can see that delta only depends on epsilon, not any other point in the interval

Question 30

cantor's theorem

Answer 30

Let f be a continuous function on a closed bounded interval ∆. Then f is uniformly continuous on ∆.

Question 31

what does it mean for a function to be Lipschitz continuous

Answer 31

∀x1, x2 ∈ ∆ : |f (x1) − f (x2)| ⩽ C|x1 − x2|.

Question 32

true or false: if a function is Lipschitz continuous

Question 33

how do we check if a function is Lipschitz continuous

Answer 33

1. If the function is differentiable

Question 34

explain how the integers are countable

Answer 34

0, 1, -1, 2, -2, 3, -3, ...., can match with 1, 2, 3, 4, 5, 6, .... as long as you order them so that they have 1-1 correspondence to the natural numbers (N).

Question 35

write down the ε - δ language for the lim of f(x) is -∞ as x tends to ∞

Answer 35

∀ ε > 0, ∃ δ s.t x > δ, you have that f(x) < -ε

Question 36

how do we prove that a function is NOT uniformly continuous on ∆ between uniform continuity and continuity

Answer 36

in normal continuity, delta changes depending on where u are along that interval for uniform continuity, delta only depends on epsillon, all points in the interval will respect these inequalities like the goodest of boys

Question 37

how do we show a function is not uniformly continuous - firstly show the definition that would be required - then tell me how

Answer 37

∣x−x′∣ < δ but ∣ f(x)−f(x′) ∣ ≥ ε. find 2 sequences xn and x\n such that xn - x\n tends to 0 but f of all that shi doesnt

Question 38

what is the easi