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f:[0,1]->|R, f(x)=x*sqrt(1+x)

Calculate Integral([0,1],f)

Calculate Integral([0,1],f)

4*(sqrt(2)+1) / 15

hint1: g:[1,sqrt(2)]->|R, g(t)=t*t-1

hint2: t=sqrt(1+x)

hint1: g:[1,sqrt(2)]->|R, g(t)=t*t-1

hint2: t=sqrt(1+x)

g : |R -> |C

h : |C -> |C

h(z) = g(Re(z))

Suppose that g is diffable. Does h have to be diffable?

h : |C -> |C

h(z) = g(Re(z))

Suppose that g is diffable. Does h have to be diffable?

NO.

Choose g(x)=x.

Then h(z)=Re(z), and h is nowhere diffable!

Choose g(x)=x.

Then h(z)=Re(z), and h is nowhere diffable!

Let A be a connected subset of |R.

Let g : A -> |R.

Suppose that g is diffable and /\x:-A g'(x)=0.

What can we say about function g?

Let g : A -> |R.

Suppose that g is diffable and /\x:-A g'(x)=0.

What can we say about function g?

It is constant.

Use Lagrange's mean value theorem.

Use Lagrange's mean value theorem.

Let X be a metric space.

Let x[n] be a sequence in X.

Let B = \\//(k=1 to oo) //\\(n=k to oo) B(x[n], 1/n).

Suppose that B is not empty.

What can we conclude?

Let x[n] be a sequence in X.

Let B = \\//(k=1 to oo) //\\(n=k to oo) B(x[n], 1/n).

Suppose that B is not empty.

What can we conclude?

sequence x[n] converges

Let X be a set, and let W c P(X).

Define W-convergence in X.

Define W-convergence in X.

Let (T,>) be a directed set. Let x:T->X be a net in X. Let k:-X.

The net x W-converges to k

iff

/\B:-W [ k:-B ==> \/t0:-T /\t>t0 x(t):-B ]

page 44 in the General Topology notebook

The net x W-converges to k

iff

/\B:-W [ k:-B ==> \/t0:-T /\t>t0 x(t):-B ]

page 44 in the General Topology notebook

Let S be a set. Give a collection of sets, W c P(|R^S), such
that pointwise convergence of functions S->|R is equivalent to
W-convergence.

If s:-S, and Uc|R is open, define U(s) = {f:-|R^S : f(s):-U}.

Let W = {U(s): s:-S and Uc|R open}.

page 44 in the General Topology notebook

Let W = {U(s): s:-S and Uc|R open}.

page 44 in the General Topology notebook

Let X be a set, and W1,W2 c P(X).

Prove that if W1 c W2, then W2-convergence implies W1- convergence.

Prove that if W1 c W2, then W2-convergence implies W1- convergence.

page 45 in gen top

Let X be a set, and W c P(X). Let d be a metric on X.

Define what it means that W is compatible with d.

Define what it means that W is compatible with d.

W-convergence is the same as d-convergence

page 46 in the Gen Top notebook

page 46 in the Gen Top notebook

Let X be a set, and W c P(X), A c X, p:-X.

Define "p is W-adherent to A".

Define "p is W-adherent to A".

\/(T,>)directed set \/a:T->A [ net a W-converges to p ]

Notice the formal difficulty here.

page 46 in the Gen Top notebook

"p is W-adherent to A" =

= "there exists a net contained in A that converges to p"

Notice the formal difficulty here.

page 46 in the Gen Top notebook

"p is W-adherent to A" =

= "there exists a net contained in A that converges to p"

Let X be a set, and W c P(X), A c X.

Define "the boundary of A" in the language of generalized convergence.

Define "the boundary of A" in the language of generalized convergence.

the set of all points which are W-adherent to A and W-adherent
to X\A

page 46 in the Gen Top notebook

Let Fr(A) denote the boundary of A.

page 46 in the Gen Top notebook

Let Fr(A) denote the boundary of A.