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, d₁ 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:

1. 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.
2. Taylor remainder theorem.

Toolkit of examples that illustrate basic ideas:

1. Examples to illustrate various aspects of the theorem 1.
1. F_n(x) = xⁿ 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.

2. 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.

2. Various examples of open/closed sets in various metric spaces. (Ex. R, R²).

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

· 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 x₀ = 0).

o Convergence with remainder theorem.