What's the other term (not continuous) that means the curve is smooth? I think that's described by f'(x) is continuous.
Differentiable? Differentiability implies continuity, yes. But that would be premature here.
(You can be continuous without being differentiable, for the record. At least SP can.)
2: isn't the term "smooth"? Good terminology!
I just saw your pointy continuous curve and I thought it's continuous but still weird but didn't know what word describes that.
https://en.wikipedia.org/wiki/Smoothness
6: It has a cusp, or it's not differentiable at that point.
No real hits on google scholar for "groovy curve." Mathematicians what are you even doing?
If it's not a Sade song, we don't want to define it precisely.
9. Out of interest, and boredom, I actually googled that, and it found the undergraduate thesis "Zombie Fever: Forecasting an Undead Epidemic". There are however many more hits for "kinky curve".
Very elegant presentation. I hope that lesson 3 is asymptotics so that the [few?] misguided students who haven't already given up all hope of understanding the course recognize that no, really, this all is incomprehensible.
Question:
On the bottom of page 5, just below the line "If all three of these agree, then the function is continuous at x=c," there are three graphs.
The second is not continuous.
If the circle at f(1) were filled in, would it still not be continuous? Because the limit as x approaches 1 from the left is slightly lower than the limit as x approaches 1 from the right?
Or, if the circle at f(1) were filled in, would it be continuous, because the limit as x approaches 1 is approximately 2 from either direction, even though it's slightly lower from the left than from the right?
Heebie, I really appreciate this!
Good question! This one is right; it would be continuous:
Or, if the circle at f(1) were filled in, would it be continuous, because the limit as x approaches 1 is approximately 2 from either direction, even though it's slightly lower from the left than from the right?
The ambiguity is because the drawing is carrying some community norms of how these diagrams tend to be drawn. The open circle is supposed to have zero width, and the sides are implied to lie on the same line.
The vocabulary word here is that it's a "removable discontinuity", meaning if that single point were filled in, it would be continuous.
If the picture had been intended to convey this:
If the circle at f(1) were filled in, would it still not be continuous? Because the limit as x approaches 1 from the left is slightly lower than the limit as x approaches 1 from the right?
then to be consistent, we'd have a left hand limit that was slightly different from the right hand limit. So the overall limit would not exist.
Shit, when I pull up that part on my phone, the three graphs are out of order in that part. I should probably go spread them out so that it's not misleading.
Not appropriate for what H-G is doing for her students here very nicely but: I always liked the asides that explain why some technique isn't obvious or why the tractable examples are special cases-- that one might be applying these definitions to functions whose properties are less clear. (Daubechies wavelets for instance; there exist artificial examples like Weierstrass function also, can't always tell what properties of an iteratively defined function are, and there are lots of those). Also higher-dimensional considerations-- existence of a derivative in one direction doesn't guarantee that all exist, so finding steepest descent might not work.
Possibly not for everyone, but seeing that there were series that failed non-obviously to converge helped motivate me personally to pay attention to convergence. Not sure there's a good way to do that in an intro without leading to confusion, but paying attention to the foundation details is worthwhile because it identifies the huge swamps where a method fails, it's not just crossing the ts to get a gold star.