Continuity, and some more Limits

Alrighty guys, my name is Justus and I will be your scribe for today. Lets get started.



This image is basically a modified version of the first slide. Now this example was used to explain the concept of continuity. If you look at the image, I've drawn 3 sets of lines, the magenta (purpley) ones, the maroony ones, and the aqua ones. If you try to travel from one aqua line, to the other, you can do it easily, and without any kind of sudden changes. Therefore the graph is said to be continuous along that interval. The same goes for the magenta. However, when trying to travel from one maroon line to another, difficulty arises. At x = 0, there is a sudden change in direction, the line is no longer continuous, and there is now said to be a discontinuity. It is called a discontinuity because there is no way to get from one maroon line to the other in a continuous fashion. The next point we got into, were 3 defining rules of continuity, which I shall type out
here for everyone to see.

Continuity

A function is continuous at x = a if;
1.) lim f(x) exists
x->a
2.) f is defined at a
3.) lim f(x) = f(a)

So after going over these rules, we got some examples on the board, so we might get a better understanding of what each means.

For the first we came up with the following

1.) Example: lim f(x) = 1/x
x->a
Counterexample: (look at slide)

So as seen in this slide image here, the example graph produces a nice, rule following reciprocal graph (reciprocal of the graph f(x) = x), whereas the counterexample gives a piecewise function, which has that huge gap/jump/aka. Discontinuity at x = 10 approximately. Alright, moving on.

2.) Example: f(x) = x/2 at x = 1
Counterexample: f(x) = 1/x at x = 0

In the second example there, it is pretty obvious to see what the problem is. In the counter example, the function is NOT defined at x = 0 because you are then dividing by zero, and we all know how much fun that is. In the example however, all is good and well, and f is defined at a, as required to be continuous.

3.) Example: lim 1/x = 1 = f(1)
x->1
Counterexample: (check slide)


Okay, so the counterexample in this one, might seem a little confusing but I'll try to explain it as best as I can. Basically, its saying that f(x) = x² when x does not equal 2; When x DOES equal 2, then f(x) = 5. Thus you get the graph with the weird point floating in the air. This point also leads to that magical thing called a discontinuity again.

Now it was at this time Dr. E said to checkout Pg's 140-141.

The final little tidbit in the lesson involved continuity on an interval, which basically says that if function is continuous from points a, and b, it is continuous on that open interval a to b. This is shown in more detail on the following slide

The lim f(x) = f(a) basically means that starting above a and moving towards it, the function
x->a⁺
is continuous. The opposite is true for the lim f(x) = f(b) which says that starting below b and x->b^-
moving towards it the function is continuous.



Now thats about it for my scribe post for today. I know we did a couple other things, (two slides), but I forgot my textbook at the school, and I didn't really understand them at all, so if someone could help out with that (ie by talking about it a little bit in their post, that would help me tremendously, and I could then go and fix mine.) Either way, like I said, I will need to get my text to help me explain those two slides.

So I think thats everything, which only leaves choosing the next scribe. I've decided it will be francis mkay. kay.

OH And I almost forget, theres alittle note about pg 143 here in my notes, which means it's probably a good idea to check that page out. As far as I remember there weren't any questions assigned for homework for tonight though.

Anyways I'm off, because I have lots of other things to do now. Hopefully my post made enough sense, and I apoligize for any discontinuities in it; feel free to point those out btw, so I may fix them.

ciao