# MA241: Real Analysis I

Unit 3: Functions and Continuity   In this unit, you will explore the properties of continuous functions. You should be familiar with continuity from calculus; indeed, many of the results which are proven in this unit will be familiar, such as the intermediate value theorem or the fact that the sum of two continuous functions is continuous. However, you will now be proving these results rigorously and with more generality. You will also be learning new concepts, such as uniform continuity, and you will be improving your knowledge of what might be called “mathematical culture” through exposure to famous and much-employed examples, such as the Dirichlet function and the Cantor Ternary set. Learn these examples well; they are used in many cases as counterexamples. Be sure also to note the use of the Triangle Inequality in the proof that the uniform limit of a sequence of continuous functions is itself continuous. This is a standard and well-known technique.

This unit should take you approximately 28 hours to complete.

☐    Subunit 3.1: 20 hours

☐    Subunit 3.2: 3.5 hours

☐    Subunit 3.3: 4.5 hours

Unit3 Learning Outcomes
Upon successful completion of this unit, you will be able to: - define continuity; - state, prove, and use properties of limits of continuous functions; - state and use properties of rational functions; - define divergence of functions to infinity and use properties of infinite limits; - define monotonicity; - prove that continuous functions attain extreme values on compact sets; - state and prove the intermediate value property; - define uniform continuity and show that given functions are or are not uniformly continuous; - give standard examples of discontinuous functions, such as the Dirichlet function; - prove that monotone functions have only a finite number of discontinuities; - define connectedness, and identify connected and disconnected sets; - construct the Cantor ternary set, and state its properties; - distinguish between pointwise and uniform convergence; - and prove that if a sequence of continuous functions converges uniformly, their limit is also continuous.

3.1 Functions   3.1.1 Basic Definitions   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 1: Basic Definitions” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 1: Basic Definitions” (PDF)

`````` Instructions: Please read the indicated section on pages 149-157.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````

3.1.2 General Theorems on Limits and Continuity   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 2: Some General Theorems on Limits and Continuity” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 2: Some General Theorems on Limits and Continuity” (PDF)

`````` Instructions: Please read “Some General Theorems on Limits and
Continuity” on pages 161-166.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````
• Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 20: Functions – Limits” Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 20: Functions – Limits” (YouTube)

Instructions: Please watch the entire video. In this lecture, Professor Su defines limits and limit points of functions. He defines continuity of a function using neighborhoods and using the sequential criterion. He states several properties of continuous functions, including the fact that the inverse image of an open set (under a continuous function) is open.

Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.

• Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 21: Continuous Functions” Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 21: Continuous Functions” (YouTube)

Instructions: Please watch this lecture, in which Professor Su discusses the definition of continuity at a point. He proves that the inverse image of an open set under a continuous function is open and illustrates this fact with several examples. He proves that the composition of continuous functions is continuous. Finally, he proves that the forward image of a compact set under a continuous function is compact and mentions some important corollaries for real-valued functions. This lecture will cover topics you have learned in sub-subunits 3.1.1-3.1.2.

Watching this lecture and pausing to take notes should take approximately 1 hour.

3.1.3 Operations on Limits   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 3: Operations on Limits and Rational Functions” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 3: Operations on Limits and Rational Functions” (PDF)

`````` Instructions: Please read the indicated section on pages 170-174.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````

3.1.4 Infinite Limits   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 4: Infinite Limits and Operations in E*” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 4: Infinite Limits and Operations in E*” (PDF)

`````` Instructions: Please read the indicated section on pages 177-180.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````

3.1.5 Monotone Functions   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 5: Monotone Functions” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 5: Monotone Functions” (PDF)

`````` Instructions: Please read the “Monotone Functions” section on pages
181-185.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````

3.1.6 Continuity on Compact Sets   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 8: Continuity on Compact Sets and Uniform Continuity” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 8: Continuity on Compact Sets and Uniform Continuity” (PDF)

`````` Instructions: Please read the indicated section on pages 194-200.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````

3.1.7 Intermediate Value Property   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 9: The Intermediate Value Property” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 9: The Intermediate Value Property” (PDF)

`````` Instructions: Please read “The Intermediate Value Property” on
pages 203-209.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````
• Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 22: Uniform Continuity” Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 22: Uniform Continuity” (YouTube)

Instructions: Please note that this video lecture covers the topics outlined in both sub-subunits 3.1.6 and 3.1.7 of this course. Please watch this lecture, in which Professor Su recaps some basic facts about continuous functions. He defines uniform continuity and relates it to compactness, giving a number of examples. He states and proves the Lebesgue Covering Lemma (this is Theorem 1 in Section 4.7 of Zakon’s book). He proves that continuous functions map connected sets to connected sets. Finally, he proves the Intermediate Value Property.

Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.

3.1.8 Discontinuous Functions   - Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 23: Discontinuous Functions” Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 23: Discontinuous Functions” (YouTube)

`````` Instructions: Please watch this lecture, which reviews material
from sub-subunit 3.1.5.  In this lecture, Professor Su discusses
some famous (or merely standard) examples of discontinuous
functions, including the Dirichlet function. He discusses right-hand
and left-hand limits of functions. He also discusses monotone
functions and why they can only have a finite number of
discontinuities.

Watching this lecture and pausing to take notes should take
approximately 1 hour.

displayed on the webpages above.
``````

3.2 Arcs, Curves, and Connected Sets   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 10: Arcs and Curves; Connected Sets” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 10: Arcs and Curves; Connected Sets” (PDF)

`````` Instructions: Please read “Arcs and Curves; Connected Sets” on
pages 211-215.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````
• Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 14: Connected Sets, Cantor Sets” Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 14: Connected Sets, Cantor Sets” (YouTube)

Instructions: Please watch this lecture, in which Professor Su states a few more results about compactness and constructs the Cantor ternary set. He defines perfect and connected sets. He proves that nonempty closed intervals are connected.

Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.

3.3 Sequences of Functions: Pointwise and Uniform Convergence (Z, L)   - Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 6, Section 1: Pointwise and Uniform Convergence” Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 6, Section 1: Pointwise and Uniform Convergence” (PDF)

`````` Instructions: Please read the indicated section on pages 189-193.