Definitions/Statements for Quiz 2

Feel free to right down any of the definitions and statements we need to know for Friday's quiz!
Here are two:

Continuity

• $f$ is continuous at $x_{0}$ if $\forall$ $\epsilon > 0$ $\exists$ $\delta >0$ such that $\vert f(x)-f(x_{0}) \vert <\epsilon$ whenever $\vert x-x_{0} \vert<\delta$.

Compact Set

• $E \subset \mathbb{R}$ is called compact if every sequence in $E$ has a subsequence that converges to a point in $E$.

Hey @ediazloa , I am sure it is probably a typo, but in Continuity I think it should be that $\lvert f(x) - f(x_0) \rvert < \epsilon$.

Here are two more:

Functional Limits

• Let $f$ be defined on an open interval $I$ and let $a \in I$. We say that $lim_{x\to a} f(x) = L$ if for every $\epsilon >0$, there is a $\delta >0$ such that $\lvert f(x) - L \rvert < \epsilon$ whenever $0<\lvert x - a \rvert < \delta$.

This next one is right out of my notes but varies slightly from the book:

Thm 4.2.3 (Sequential Characterization of Functional Limits)

• Suppose $f$ if defined on a set $A$, which has $a$ as a limit point. Then $lim_{x \to a} f(x) = L $ iff $lim_{n \to \infty} f(x_n) = L$ for all sequences $(x_n)$ in $A$ with $x_n \neq a$, but $lim_{n \to \infty} x_n = a$.

The book version follows:

Thm 4.2.3 (Sequential Criterion for Functional Limits)

Given a function $f : A \mapsto \mathbb R$ and a limit point $c$ of $A$, the following two statements are equivalent:

$$ \text{(i)} \:lim_{x \to c} f(x) = L.$$
$$\text{(ii) For all sequences} \: (x_n) \subseteq A\: \text{satisfying} \:x_n \neq c\: \text{and} \:(x_n) \rightarrow c, \text{it follows that} \\f(x_n) \rightarrow L.$$

Thanks for catching that typo!!

@ediazloa another typo, for compact set you want $E\subseteq \mathbb{R}$ is called compact if...

wow... thanks! I don't know what was up with that.

Heine-Borel Theorem: A set $K\subseteq\mathbb{R}$ is compact if and only if it is closed and bounded

Closed: a set $F\subseteq\mathbb{R}$ is closed if it contains its limit points.

Hey @agibson, I think you have a good definition for bounded as the Heine-Borel Thm.

The Thm. actually states that:
A set $A \subseteq \mathbb{R} $ is compact if and only if it is closed and bounded.

You are so right! I was looking at the definition above it and obviously wasn't paying as much attention as I needed to. My bad... thanks for catching that!

a function, $f$, is unifromly continuous if $\forall$ $\epsilon > 0$ we can pick a $\delta$ s.t.
$\left|f(x) - f(x_{0})\right| < \epsilon$ whenever $\left|x - x_{0}\right| < \delta$ and $\delta$ is not dependent on the value of $x_{0}$.

So, is this the right list for definitions:
open
closed
compact
Heine-Borel Theorem
functional limit
sequential characterization of functional limit
continuity
limit characterization of continuity

Am I missing any?

That is the list I have that he put on the board and took a picture of. I am just not sure about open and closed. Do you think we should include what exactly a limit point is and exactly what an interior point is?

I personally don't think we necessarily need to write it down in the definitions, but it could be helpful to at least be comfortable with. Also, for open I used the definition from the book that states, "a set $O\subseteq\mathbb{R}$ is open if for all points $a\in O$ there exists an $\epsilon$-neighborhood $V_{\epsilon}(a)\subseteq O$."

I guess I wrote the following on the board one day, though, I also added uniform continuity later.

IMG_0167.jpg3264x2448

In the book, Thm 4.2.3 is only labelled as the Sequential Criterion for Function Limits. Is there something I am missing where it has both definitions under one thm?

The book says:
A function is uniformly continuous on A if for every $\epsilon$>0 $\exists \delta$ s.t. $|x-y|<\delta$ implies $|f(x)-f(y)|<\epsilon$.
I interpret your $x_0$ to be a constant which isn't what I'm getting from the book.

Yes. The first one I listed is from my notes. And the second is straight out of the book. I tried to indicate that but sorry if it is not clear.

For limit characterization of continuity do we just need to know: f is continuous at a if $lim_{x\rightarrow a}f(x)=f(a)$?

@agibson The only additional details I have are that if $f$ maps some set $A$ to the real numbers, $a$ is a limit point of $A$ and $a\in A$.

$x_0$ is the same thing as $y$ in your definition. It is an arbitrary point in the domain of our function $f$, a function is uniformly continuous if we may pick our $\delta$ without regard to this point. e.g. $f(x) = ax + b$ with $a,b \in \mathbb{R}$, for which our $$\delta = \frac{\epsilon}{\left|a\right|}.$$

@jmacdona I think this is just a typo but you would probably choose $\displaystyle \delta = \frac{\varepsilon}{\left| a \right| }$.