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 fn 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. Fn(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.

 

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

                                                                      i.       Fn(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. fn(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.

 

  1. 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: ex, sin x, cos x. (note here x0 = 0).

o        Convergence with remainder theorem.