These notes are also on reserve in paper
form at the Kersey Library.Here are some of the lecture notes from the MATH 501: Real
Analysis I class. To read them online you will need Adobe Acrobat
Reader version 3.0 or newer. This is available free from Adobe. These notes are also on reserve in paper
form at the Kersey Library.
Title |
Description |
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1. |
Set Theory (Last modified 9/18/98.) |
Basic notation for sets, relations, functions, cardinality. |
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2. |
The Axioms for the Real Numbers (Last modified 9/18/98.) |
Field, order, completeness axioms. Metrics. Existence of an irrational number. |
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3. |
Sequences (Last modified 10/4/98.) |
Convergence, boundedness, Sandwich Theorem, monotone sequences, Nested Interval Theorem, subsequences. |
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4. |
The Topology of R (Last modified 10/10/98.) |
Open and closed sets, limit points, Bolzano-Weierstrass theorem. |
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5. |
Cauchy Sequences (Last modified 11/6/98.) |
Characterization of convergence for a sequence. |
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6. |
Covering Properties and Compactness
on R (Last modified 11/6/98.) |
Open covers, Lindelöf property, compactness. |
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7. |
Connectedness (Last modified 11/6/98.) |
A set in R is connected iff it is a point or an interval. |
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8. |
Limits of Functions (Last modified 11/6/98.) |
Definition of limit, relation to sequences, sum, product, quotient, Squeeze Theorem. |
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9. |
Unilateral Limits (Last modified 10/12/98.) |
Application to monotone functions. |
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10. |
Continuity (Last modified 11/6/98.) |
Various equivalent definitions, sum, product, quotient, composition. |
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11. |
Unilateral Continuity (Last modified 10/12/98. More to come.) |
Jump discontinuity, removable discontinuity, monotone functions are "nearly everywhere" continuous. |
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12. |
Continuous Functions (Last modified 10/12/98. More to come.) |
Topological equivalence in terms of open sets, continuous image of a compact set is compact, inverse is continuous, continuous image of a connected set is connected, Darboux property. |
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13. |
Uniform Continuity (Last modified 10/12/98. More to come.) |
Continuous on compact set implies uniformly continuous. |
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14. |
Differentiation (Last modified 10/12/98. More to come.) |
Sum, product, quotient and chain rules, derivative of the inverse of a function, extrema. |
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15. |
Differentiable Functions (Last modified 10/12/98. More to come.) |
Mean value theorems, monotonicity, Darboux property. |
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16. |
Applications of the Mean Value Theorems (Last modified 10/12/98. More to come.) |
L'Hôspital's rules, Taylor's theorem. |
| Bibliography (Last modified 10/12/98.) |
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