Analysis Term 2

If f is a power series then it has a continuous derivative over...

(-R,R)

Integrability of function which is discontinuous at finitely many points

(ignore discontinuities in partition)

Proof of Cauchy's mean value theorem

- Define new function h(x) which is a linear combination of f(x) and g(x) - Show that h(a)=h(b) - Apply Rolle's theorem

Rewriting defintion of convex function

- Write t as a wighted average of c and d since it lies between them: let t=sc+(1-s)d for some s∈[0,1]

A power series converges __________ anywhere inside radius of convergence

A power series converges absolutely anywhere inside radius of convergence

Idea of proof for mean value theorem

Add/subtract linear terms from the function, f, so that the red lines become horizontal, then apply Rolle's theorem

Does a set have to be either open of closed?

No can be both or neither

Mean value theorem

Note that we require the function to be continuous at the end points but not differentiable

Link between Taylor's theorem and MVT

When we take n=0 in Taylor's theorem, we get the MVT

How can we write the set S={y>0|sin(y)=0} in terms of π?

S={nπ|n∈N}

Is the empty set open or closed?

Both (vacuously)

Is R closed? Is it open? It is compact?

Closed and also open, but not compact

Definition of uniform continuity

Given an epsilon, and have to choose a delta which works with all a at the same time

Continuous functions send a closed and bounded interval to....

another closed and bounded interval

Uniform continuity depends on...

the function itself but also on the domain

Definition of limit

x close to a (but not equal to) implies f(x) close to b

Proof that if f'(x)=g'(x) then there exists some c∈R such that f(x)=g(x)+c

- Define h(x)=f(x)-g(x) - Differentiate - Use result that derivative of 0 means the function is constant

Idea of proof that composing an integrable and continuous function gives integrable function

- Find partition, P, with U(h,P)-L(h,P)<ε - split intervals of P into two types, either (i) f doesn't change much (so neither does gf by uniform continuity of g), or (ii) f changes a lot, which is rare - this will give us bounds on U(h,P)-L(h,P)

How to show that f is NOT uniformly continuous

- Fix epsilon - Choose x and y depending on delta, such that |x-y|<delta is satisfied, but |f(x)-f(y)|≥epsilon i.e find sequences x_n and y_n where |s_n-y_n| converges to 0 (THIS DOES NOT MEAN THAT THE CONVERGE TO THE SAME THING) but |f(x_n)-f(y_n)| does not? -Convergent sequences will not work so think about 1/n, 1/(n+1) etc...

Proof that there exists y>0 s.t. sin(y)>0

- Suppose not - Then cos(x)≠0 ∀x>0 also - By IVT, we have cos(x)>0 ∀x>0 - This means derivative of sin is positive and so sin is strictly monotone increasing, and sin(x)>0 ∀x>0 also - Apply MVT to cos between (1,x) for any x>1 - Rearrange for contradiction

Second derivative test proof (minimum case)

- Use limit definition of derivative - Since we assume f''(a)>0, we have that the expression inside limit >0 also within some δ neighbourhood of a - Then can show f is strictly decreasing on first bit of interval, and strictly decreasing on the second - Therefore, local minimum

Proof that sin and cos are periodic functions

- We know π is first root of sin so sin(π/2)>0 and cos(x)=1-2sin²(π/2)<1 - Can deduce cos(2π)=0 - Use double angle formulae to show sin(x+2π)=sin(x) and cos(x+2π)=cos(x) - Suppose there is a smaller period and come to contradiction

Idea of proof that we require uniform convergence for integral of limit to equal limit of integrals

- fₙ is very close to f when n is large - so upper and lower darboux sums for f must be very close to those for fₙ - so the upper and lower darboux integrals get arbitrarily close to the integral of fₙ as n→∞

Idea of proof of integration by parts

- start with product rule - f' and g' are continuous so f'g and fg' are continuous - and so (fg)'=f'g+fg' is integrable - apply 2nd FTC

Sine and cos are periodic with period of...

2π

Alternative graphical interpretation of convex (regions)

A line joining any two points above the function doesn't cross below the function

A sequence of continuous functions needs to be ______ in order to converge to something continuous

A sequence of continuous functions needs to be uniformly continuous in order to converge to something continuous

All ________ functions are integrable

All continuous functions are integrable

Image of continuous function on closed bounded interval

Also a closed bounded interval

Proof that f'(x)=0 implies constant

Alternative proof: if x<y in [a,b], then by MVT we know ∃c∈(x,y) s.t: (f(y)-f(x))/(y-x)=f'(c)=0. This implies f(y)=f(x)

Definition of bounded

Both bounded above and bounded below

What do we know about the set {L(f,P)|P is a partition of [a,b]}

Bounded above by any UD sum, U(f,Q) (any Q works)

What do we know about the set {U(f,P)|P is a partition of [a,b]}

Bounded below by any LD sum, L(f,Q) (any Q works)

Why are sine and cosine continuous?

Can be defined in terms of e which we know is continuous

Approximating sin using Taylor's Theorem

Can find upper bound for remainder term

The stringer version of the mean value theorem is called...

Cauchy's mean value theorem

is [13,∞) closed? Is it compact?

Closed but not compact

Useful tool for proving things about two functions

Define a new function as the sum/difference of them

Definition of continuity everywhere

Delta can depend on a (but not always necessary)

What requirement do we not require for Taylor's theorem?

Don't need (n+1)th derivative to be differentiable

If f is strictly increasing, then the supremum is located where on the interval?

End

How to show a function has a root using IVT?

Find a point ≤ 0 and another point ≥ 0. Then apply IVT

How can we show a function is integrable, without knowing the integral

Find upper bound for the integrand which is integrable

Definition of open set

For all x in S, the delta neighbourhood of x is also in S

Useful think to thing about when asked for example of convergent sequences with discontinuities

For x∈(0,1), xⁿ→0, but 1ⁿ→1

Graphical interpretation of convex (lines)

Function lies below line joining the two end points

Sequential continuity

Given a sequence that lies completely in the domain of a function with a limit of a, f(x_n) will have a limit of f(a)

Link between analytic functions and Taylor series

If a function is equal to its own Taylor series, then we say that it is analytic

If a sequence of continuous functions converges uniformly, then they must converge to a _______ function

If a sequence of continuous functions converges uniformly, then they must converge to a continuous function

When is the second derivative test inconclusive?

If f(x) satisfies f'(a)=f''(a)=0. Then it does not tell us whether f has a local maximum or local minimum at x=a.

When can we differentiate a sum term by term?

If it is finite, or if we have proven convergence

First fundamental theorem of calculus

If you integrate a function that is continuous, then (i) the integral is a continuous function (ii) its derivative exists and is the original function

Integration is a ______ operator

Integration is a linear operator

We say a function f:[a,b]→R is convex is for all c<t<d in the domain, we have...

LHS is a linear function in t which passes through (c,f(c)) at t=0, and (d,f(d)) at t=d.

If we have a sequence of functions that converge pointwise, what can we say about the limits of the integrals

Limit of integrals not necessarily equal to integral of limits

Definition of a closed set

Limits of all convergent subsequences in set are also in set

how do we check that f is continuous everywhere?

Must satisfy condition of continuity for all a. This is difficult to check so we use sequential continuity instead

Can we use the MVT to prove L'Hoptial?

NO since L'Hopital doesn't require g to be a continuous function

For a Taylor series of P(x) of f(x) about a=0, is it always true that we have f(x)=P(x) on (-1,1)?

No, consider 'bump function' as counterexample

Does the Taylor series of a function at a=0, P(x), have an infinite radius of convergence?

No, consider f(x)=1/(1-x) as counterexample

Is every function equal to its own Taylor series at a point?

No, consider this counterexample where f(x)=P(x) only when x=0

Are all integrable functions continuous?

No, counterexample:

When is a power series is integrable?

On a closed subinterval of [-R,R] where R is radius of convergence

If a set S is open, then what can we say about its compliment?

R\S is closed

Taylor's Series definition

Requires function to be infinitely differentiable

The mean value theorem is a more general case of...

Rolle's theorem where we do not necessarily have f(a)=f(b)

Sequences converge _______. Functions converge _______.

Sequences converge absolutely. Functions converge uniformly.

Using IVT how can we prove there exists more than one x s.t f(x)=c?

Split the function up and apply IVT to the different sections

If f is strictly increasing, then the infimum is located where on the interval?

Start

Definition of partition

Strictly increasing sequence of real numbers

Cauchy's mean value theorem

Stronger than mean value theorem

Taylor polynomial for a function which has its (n+1)th derivative equal to 0 at all x for some n

Taylors theorem says that f(x)=P_n(x) and so f must be a polynomial of degree at most n

How to integrate power series

Term by term

What do we require to define the infimum and supremum over an interval?

That the function is bounded on that interval (but doesn't have to be continuous!)

What requirements do we need in order for nth derivative of f to exist at x=a?

The (n-1)th derivative must exist in a neighbourhood of a and be differentiable at x=a

What is the partition of an interval?

The partition splits up an interval into closed intervals

What condition do we require for the limit of the derivatives of a sequence of functions to converge to the derivative of the limit of the functions?

The sequence of the derivatives of the functions converge uniformly. Note: we only need the sequence of functions to converge pointwise

What function do we define in order to prove the mean value theorem?

Then differentiate and apply Rolle's theorem

Example of function where neither upper or lower Darboux sum exist

Unbounded on every interval

What condition do we need on a sequence of functions, fₙ, for the limit of the integral to be the integral of the limit?

Uniform continuity

Proof that cos²x+sin²x=1

Use difference of two squares and the fact that cos is even and sin is odd

Proof of L'Hopital's rule from one-sided version

We have proven L'Hopital for x↑a and analogously for x↓a so we can just combine these

If a function is differentiable and we know the derivative, then what do we know about the function?

We know the function up to some additive constant

When you see the sum of a family of functions, think about...

Weierstrass M-Test

What is an improper integral?

When the integral is not defined at some point

Idea of proof that the product of two integrable functions is integrable

Write fg as a linear combination/composition of integrable/continuous functions

Is f(x)=x convex?

Yes

Integrability of monotone increasing functions

always integrable

Integrability of strictly monotone increasing function, f (inverse)

both f and f⁻¹ are integrable

The intersection of (possibly infinitely many) closed sets is...

closed

The union of finitely many closed sets is...

closed

Definition of compact

closed and bounded

NOT SURE IF THIS IS TRUE: If the upper and lower darboux sums are equal on all partitions then the function is...

constant

What condition is not required for L'Hopital?

continuity

If f is continuous and injective, the f⁻¹ is...

continuous

f'(x) is integrable if it is...

continuous

If f is differentiable at a, then it is also...

continuous at a

Define cos in terms of the real part of another function

cos(x)=Re(E(ix))

L'Hopital's Rule

don't actually understand!!

If a function is constant then its upper and lower darboux sums are...

equal over any partition

If we bound something between 0 and ε, then it must be...

equal to 0

Sequential compactness: A subset of R, S, is compact iff...

every sequence which is a subset of S has a convergent subsequence which converges to a limit in S

For a bounded function, how do we know and inf and sup exist over an interval?

extreme value theorem

Three equivalent statements about a continuous f:[a,b]→R

f injective ↔f strictly monotonic ↔f is a bijection

If a sequence of (uniformly) continuous functions f_n:S→R converges uniformly to f, then...

f is (uniformly) continuous

If f:[a,b]→R is continuous (on closed bounded invterval), then...

f is bounded

For f continuous on [a,b] and differentiable on (a,b). If f'(x)=0 for all x∈(a,b), then...

f is constant on [a,b]

Definition of continuity at point a

f is continuous at a point a, if and only if for all epsilon there exists a delta such that for all x in a delta neighbourhood of a, f(x) is in an epsilon neighbourhood of f(a)

If f⁻¹ is integrable then...

f is integrable

For f continuous on [a,b] and differentiable on (a,b). If f'(x)≥0 then...

f is monotone increasing on [a,b]

For f continuous on [a,b] and differentiable on (a,b). If f'(x)>0 then...

f is strictly monotone increasing on [a,b]

If f'(x) is monotone increasing then...

f''(x)≥0

If the lower and upper Darboux integrals are not equal, then...

f(x) is not integrable on [a,b]

If f and g are continuous at a, then the following are also continuous at a... What do we use to prove these?

f+g, f-g, fg, f/g (iff g(a)≠0) Proven by sequential continuity and algebra of limits

If strictly monotonic increasing then...

for any distinct points x<y in the domain we have f(x)<f(y)

If strictly monotonic decreasing then...

for any distinct points x<y in the domain we have f(x)>f(y)

The original MVT is the Cauchy MVT with...

g(x)=x

A bounded function f:[a,b]→R is integrable iff...

i.e: as long as we can get lower and upper sums arbitrarily close for a subset of partitions, then integrable

If f:[a,b]→R or f:R→R is continuous, the f is injective

iff f is strictly monotonic

If f:[a,b]→R is a continuous function, then f is...

integrable

If we compose an integrable (not necessarily continuous) function with a continuous function then the result is...

integrable

Linear combinations of integrable functions are...

integrable

The product of two integrable functions is...

integrable

The antiderivative of f(x) is the...

integral of f(x)

If f is bijective then f has an...

inverse

f:[a,b]→R continuous means that...

it is uniformly continuous since we are going from a compact set

f'(a)=

lim (f(x)-f(a))/(x-a) or...

What inequality do we always know is true for upper and lower darboux sums

lower≤upper

Rolle's theorem is a special case of the...

mean value theorem, where f(a)=f(b)

A half open interval is...

neither open nor closed

Any lower Darboux sum for f is ≤ any upper Darboux sum of f, regardless...

of the partitions used

The intersection of a finite number of open sets is...

open

The preimage of an open set is...

open

The union of of open subsets (possibly infinitely many) of R is...

open

Integral test for convergence of a series

ps7

Define sin in terms of the imaginary part of another function

sin(x)=Im(E(ix))

If f is convex on (x,y), and we take some c,d∈(x,y), then what can we say about the slopes between various points?

slope between x and x ≤ slope between x and y ≤ slope between d and y

What do we do when finding an integral from a to b where f is unbounded at some interior point c∈(a,b) or where both a=-∞ and b=∞

split the integral!

Why/when is tan continuous?

tan=sin/cos and we know sin and cos are continuous. This is only when cos(x)≠0, x≠(2k+1)π/2

If S is compact and f:S→R is continuous, then f is...

uniformly continuous

If the upper and lower Darboux sums integrals of f on [a,b] are equal, then...

we say f is (Darboux) integrable on [a,b] and define: