### Liveblogging Algebra Class: Week One

I’m proud to’ve finally been able to motivate myself to post regularly; having had some students comment (here and ISCRL …hover the mouse over the acronym … isn’t that a cool trick? … oh, you’ve seen it) is very gratifying of course. Heck, there might even be some math up here that some strangers can use somewhere down the line. Oh how I love this World Wide Web. Of course, sometimes late at night I’ll get this mortal terror: what if I’m about to make more of a public fool of myself than even I’ve ever done? What if I already have? Probably it goes with the territory.

I’ll be taking off (for the whole weekend) in about half an hour here, so I’d better not start anything substantial. I’m posting whatever’s on the screen at 4:55. Here’s a little math.

For this entire section, by “function”, we will mean a Real-valued function of a Real variable.

Now, how hard was that? And don’t I wish I’d said earlier! In a little while, we’ll be talking about other functions (in considerable detail; I’ve mentioned some in passing already in class)—Complex-valued functions of a Complex variable in particular. The thing to do is to state your assumptions as clearly as you know how.

Anyhow, let f denote a function; let S be a subset of the Reals ($S\subset{\Bbb R}$; note that the “full set”—$\Bbb R$ itself— is one possibility). Then f is said to be increasing on S whenever
$[x_1 < x_2] \rightarrow [f(x_1) < f(x_2)] (\forall x_1, x_2 \in S)$.
(“Bigger x’s get bigger y’s”.) Part of the point here is that S is arbitrary (if we’re very careful, maybe we should’ve assumed that S actually has two distinct elements …). The text gives essentially the same definition, except that S has to be an interval. This way madness lies. There sure as heck are increasing functions with $\Bbb Z$ (the integers) as a domain, for example. The rest will have to wait.