First In-Class Exam Solutions: Math 412
Section 0101, Professor Levermore
Thursday, 8 March 2012
n
m
1. [15] Let f : R R be continuous. Let G Rm be open. Show that f 1 (G) is open,
where
f 1 (G) = x Rn : f (x) G .
Solution. In order to show that f 1 (G) i
Advanced Calculus: MATH 410
Real Numbers
Professor David Levermore
7 January 2012
1. Real Number System
1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly
familiar with them. Some basic sets of numbers are:
natural numbe
Advanced Calculus: MATH 410
Functions and Regularity
Professor David Levermore
7 January 2012
5. Functions, Continuity, and Limits
5.1. Functions. We now turn our attention to the study of real-valued functions that are
dened over arbitrary nonempty subse
Advanced Calculus: MATH 410
Riemann Integrals and Integrability
Professor David Levermore
7 January 2012
9. Riemann Integrals
We now revisit the denite integral that was introduced to you when you rst studied calculus.
You undoubtedly learned that the den