Definitions for Midterm:

(try to associate a picture with the definition)

- convergence:

o limit of sequence of real numbers

o pointwise convergence of sequences of functions

o uniform convergence of sequences of functions

- continuity of a function (for both R and for a metric space).

- metric space with distance function satisfying the 4 axioms.

- Examples of metric spaces.

o C[a,b] or C[0,1], d_infinity = sup metric.

o
(Discrete metric, taxi-cab, d_{1 }metric =
integral metric)

- Complete metric space

- Open and closed sets

- Cauchy sequence

- Taylor polynomial formula

- (Remainder formula for Taylor polynomials – not definition but theorem).

Major Theorems:

- If
a sequence of continuous
functions f
_{n}^{ }converge uniformly to a function f on a set A, then the limit function f(x) is continuous on the set A. - Taylor remainder theorem.

Toolkit of examples that illustrate basic ideas:

- Examples to illustrate various aspects of the theorem 1.
- F
_{n}(x) = x^{n}for x in [0,1]. Pointwise convergence to f(x) = 0 if x in [0, 1), and = 1 if x =1.

Limit function is discontinous, yet we have a sequence of continuous functions (converging pointwise). Since the limit function is not continous, the convergence can not be uniform.

- sequence of functions that do converge uniformly on a set (so the limit function is continous).

i.
F_{n}(x) = x + (sin x)/n converged uniformly to f(x) =
x for all real numbers x. Then it also converges uniformly for all x in [0,1]
or in [a, b].

(Make up your own examples. Requires a strong understanding to do this).

c.
f_{n}(x) = x/(x + n). This convereges pointwise to f(x) = 0 (note-f is
continous) for all real numbers x
yet the convergence was non-uniform. This example does not contradict the
theorem.

- Various
examples of open/closed sets in various metric spaces. (Ex. R, R
^{2}).

Main Topics:

- Metric spaces

o Distance function

¤ Triangle inequality

¤ Calculating the distance for examples.

o Cauchy sequence -> complete metric space

o Open/closed sets.

o Contiuous function:

¤ Real function defintion with absolute value = distance

á Delta – epsilon cheer.

¤ Generalize to definition with a metric d.

- sequences:

o of real numbers and limits

o of functions;

¤ pointwise

¤ uniform convergence.

- Taylor polynomials,

o
Toolkit; basic example: e^{x}, sin x, cos x.
(note here x_{0} = 0).

o Convergence with remainder theorem.

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