Continuity summary

  1. Real numbers as ordered set with multiplication and addition

  2. as complete ordered field
    1. Definition of algebraic field
    2. Definition of ordered field. Completeness axiom
    3. Definition of £
    4. Definition of upper bound
    5. Definition of ``bounded above''
    6. Definition of supremum
    7. Uniqueness of supremum
    8. Archimedean property. Use of completeness to prove existence of 2
    9. Another definition of supremum, in therms of e
    10. Supremum of unions of sets
    11. Supremum of sums of sets
    12. Definition of infimum
    13. Proof of existence of infimum on bounded-below sets, by completeness


  3. Revision on limits of functions
    1. Definition of limits of functions. One-sided limits
    2. Definition of continuity at a point
    3. Same again
    4. Definition of continuity on an interval
    5. Well-behavedness of conitnuity under addition etc.
    6. Continuity of polynomials
    7. Continuity under composition
    8. Differentiability implies continuity


  4. Results about sequences proved using completeness, and needed to study continuity
    1. Any monotone bounded sequence converges
    2. Bolzano-Weierstrass theorem (two proofs)
    3. Any sequence of has a monotonic subsequence


  5. Results about continuous functions on a bounded interval
    1. Any CFOABI is bounded and attains its bounds
    2. Special case of IVT (two proofs)
    3. IVT
    4. Corollary: existence of positive roots of positive reals
    5. Corollary: any odd real polynomial has ³ 1 real root
    6. Corollary: continuous image of a CBI is a CBI
    7. Existence of a fixed point of f:[a,b] [a,b]


  6. An inverse function theorem

  7. Intervals in
    1. Definition of intervals in terms of betweenness


  8. Integration
    1. Definition of step function
    2. Definition of partition
    3. Definition of integral of step functions
    4. Equality of integrals of subordinate partitions of a step function
    5. Definition of refinement
    6. Existence of common refinement of partitions; independence of integral from partition used
    7. Set of step functions as a vector space; integration as a linear function from this space to
    8. Positivity
    9. Biggerness of integral of a bigger function. Indicator functions


  9. Integrals of continuous functions
    1. Integral as supremum of set of all integrals of step functions
    2. Proof
    3. Corollary: Integral lies between smallest and biggest rectangles
    4. Corollary: Integral has same area as some f(x)-height rectangle
    5. How not to integrate
    6. Cobbling step functions together
    7. Integral of sum of step functions is sum of integrals
    8. Similarly for integrals
    9. Definition of upside-down integral
    10. Corollary to 9.8: addition of integrals in any order


  10. Fundamental theorem of calculus
    1. FTC mk. I (on indefinite integrals)
    2. Definition of an anti-derivative (primitive)
    3. Constant difference of anti-derivatives
    4. FTC mk. II (on anti-derivative)


  11. Applications of FTC
    1. Linearity of integral on continuous functions
    2. Corollary: biggerness of integrals of bigger functions
    3. Integration by substitution
    4. Misuse and correct use of substitution
    5. Integration by parts
    6. Alternative definition of log
    7. Equivalence of definitions of log


  12. Taylor's theorem revisited
    1. Taylor's theorem in integral form
    2. Example: power series expansion of log(1+x)


  13. Second MVT for integrals

  14. Numerical integration; error analysis
    1. Maximum error of trapezium integration
    2. Proof


  15. Metric spaces
    1. Definition of continuity for any f: n
    2. Definition of a metric space
    3. Continuity at a point for any function between metric spaces
    4. Likewise over subset of space
    5. Definition of open spherical neighbourhood
    6. Translation
    7. Definition of an open set
    8. Openness of open spherical neighbourhoods
    9. Warning: dependence of openness on subspace
    10. Continuity in terms of open sets
    11. Openness of intersections and unions of open sets
    12. Topological equivalence of metrics
    13. Transitivity of openness
    14. Definition of closedness
    15. Warning: a set can be neither open nor closed (or both)
    16. Definition of a limit point
    17. Definition of closure


  16. Compact spaces
    1. Definition of cover and subcover
    2. Definition of compactness
    3. Heine-Borel theorem (compactness of closed intervals)
    4. Boundedness and closedness of compact subspaces
    5. Converse
    6. Compactness of continuous images of compact sets
    7. Corollary: Boundedness of a continuous function on a compact space


  17. Connected sets
    1. Definition of connectedness and disconnectedness
    2. Partition of a metric space; connectedness as unpartitionability
    3. Equivalence of definitions
    4. Connectedness of open intervals in
    5. Connectedness of continuous image of connected space
    6. The image of a -valued continuous function on a connected space is an interval.
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