- #1

DrWahoo

- 53

- 0

Remember to use the appropriate packages; these are in similar post if a mod wants to add the link if you choose to use Latex.

Here is the PDF View attachment 7596

Here is the PDF View attachment 7596

Code:

```
\begin{document}
\begin{center}
{\LARGE Epsilon-Delta Proofs \\[0.25em] Practice} \\[1em]
{\large Just for practice, don't use Google to cheat!}
\end{center}
\bigskip
\begin{problem}[1.] Use an $\epsilon$-$\delta$ proof to prove
$\ds \lim_{x \to 2} \, \frac{1}{2} x = 1$.
\end{problem}
\begin{problem}[2.] Use an $\epsilon$-$\delta$ proof to prove
$\ds \lim_{x \to -2} \, (-3x + 1) = 7$.
\end{problem}
\begin{problem}[3.] Use an $\epsilon$-$\delta$ proof to prove
$\ds \lim_{x \to 3} \, (6x - x^2) = 9$.
\end{problem}
\begin{problem}[4.] Consider the function $f:\RR \to \RR$ defined as
\[
f(x) = \begin{cases} 1 & \text{if $x$ is rational} \\
0 & \text{if $x$ is irrational.} \end{cases}
\]
At which values of $x$ is $f(x)$ continuous? You can assume without proof
the fact that between every two rationals, there is an irrational and between every two
irrationals, there is a rational.
\end{problem}
\begin{problem}[5.] Give an example of a function $f$ such that $f$ is continuous nowhere, but $|f|$
is continuous everywhere. \emph{Hint:} Think about problem 4.
\end{problem}\end{document}
```