**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.*

**Unit 3 Time Advisory**

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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
```

**Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 4: Problems on Limits and Operations in E*”**Link: The Trillia Group: Elias Zakon’s*Mathematical Analysis I*: “Chapter 4: Function Limits and Continuity: Section 4: Problems on Limits and Operations in E*” (PDF)Instructions: Please scroll down to page 181, and work through problem 4.

Terms of Use:

*Mathematical Analysis I*was written by Elias Zakon and relicensed under a Creative Commons-By Attribution 3.0 Unported Licencse as part of 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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**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.
Terms of Use: Please respect the copyright and terms of use
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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**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.
Terms of Use: The article above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Jiri Lebl and the original version can be found
[here](http://www.jirka.org/ra/realanal.pdf).
```

**Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 25: Taylor’s Theorem, Sequences of Functions”**Link: YouTube: Harvey Mudd College: Professor Francis Su’s*Real Analysis*: “Lecture 25: Taylor’s Theorem, Sequences of Functions” (YouTube)Instructions: Please play the video from the 36-minute mark to the end. In this lecture, Professor Su discusses sequences of functions and what it means for them to converge either pointwise or uniformly. He gives several classic examples of sequences of functions which converge pointwise to zero. He proves that if a sequence of continuous functions converge uniformly, their limit is continuous.

Watching this lecture and pausing to take notes should take approximately 45 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.