## ~sbinet/star-tex

star-tex/testdata/hello.tex -rw-r--r-- 807 bytes
9bb27e8aSebastien Binet ci: bump to freebsd/latest 7 months ago

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
The foundations of the rigorous study of {\it analysis}
were laid in the nineteenth century, notably by the
mathematicians Cauchy and Weierstrass. Central to the
study of this subject are the formal definitions of
{\it limits} and {\it continuity}.

Let $D$ be a subset of $\bf R$ and let
$f \colon D \to {\bf R}$ be a real-valued function on
$D$. The function $f$ is said to be {\it continuous} on
$D$ if, for all $\epsilon > 0$ and for all $x \in D$,
there exists some $\delta > 0$ (which may depend on $x$)
such that if $y \in D$ satisfies
$$|y - x| < \delta$$
then
$$|f(y) - f(x)| < \epsilon.$$

One may readily verify that if $f$ and $g$ are continuous
functions on $D$ then the functions $f+g$, $f-g$ and
$f.g$ are continuous. If in addition $g$ is everywhere
non-zero then $f/g$ is continuous.
\bye