Definitions for Midterm:
(try to associate a picture with the definition)
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).
Toolkit of examples that illustrate basic ideas:
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.
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.
- 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.
o of real numbers and limits
o of functions;
¤ uniform convergence.
- Taylor polynomials,
o Toolkit; basic example: ex, sin x, cos x. (note here x0 = 0).
o Convergence with remainder theorem.