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sniffnoyBlech. I've been tired enough the last few days that I've actually ended up napping once or twice. Mark found me sleeping in the lounge the other day and suggested I go to bed early, which I proceeded to do. Although it seems that so far this has only happened when working on Hum. :P
So Boy Alex stayed up all night last night, and when I came back to the room at only about 17:00 or so today, he was in bed. I figured he was just taking a nap, but I came back again later, and later, and it seems nope, he'd gone to bed. Now I have to wonder if he's going to wake up at 9:00 or if he's going to wake up at 6:00.
So for some reason Girl Alex, Shuo, and Nadja got the idea to try acting like *nuns* for the week. Alex seems to be one who's been all "Now I'm in college and I'll try all sorts of new things," and... well... this is where that led. It has... failed miserably. It wasn't too bad before, but now Alex has apparently just gone out and gotten drunk.
So today a trampoline appeared in our lounge. Noone was really sure where it came from. Some people said Henderson had a trampoline, and we must have stolen it from them, while others said Shorey had a trampoline, and we stole it from them. Eventually it came out that some guy named Noah from Thompson stole it from Shorey and gave it to us, because Shorey was always jumping on it and annoying the Thompson kids below them, so they gave it to us because there's no one below us. Only it turns out, Shorey actually originally stole the trampoline from Henderson so that they could make slam dunks in their basketball hoop.
Neat little problem we got a bit ago in math:
(I should note that we have not talked about Lebesgue measure yet, I assume we won't for quite a while, but we have defined measure 0 for subsets of R.)
So consider the function f(x)=1/denom(x) for x∈Q, 0 otherwise. Only modify it so that f(0)=0, and actually we're going to consider fr (exponentiation, not iteration) for r≥1. As you know, fr is continuous precisely on the irrationals and 0.
Then:
1. Show fr not differentiable on the irrationals for r≤2.
2. For what r is f differentiable at 0?
3. (The neat part!) Show that fr is differentiable almost everywhere for r>2.
(Hint: First show that A⊆R is measure 0 iff you can cover it with a countable collection of open intervals such that each element of A is in infinitely many elements of this cover, and so that the total length of the cover is finite.)
I think I'll start referring to people here using their house in place of their last name when this is unambiguous, e.g. "Lisa of Flint".
-Sniffnoy
So Boy Alex stayed up all night last night, and when I came back to the room at only about 17:00 or so today, he was in bed. I figured he was just taking a nap, but I came back again later, and later, and it seems nope, he'd gone to bed. Now I have to wonder if he's going to wake up at 9:00 or if he's going to wake up at 6:00.
So for some reason Girl Alex, Shuo, and Nadja got the idea to try acting like *nuns* for the week. Alex seems to be one who's been all "Now I'm in college and I'll try all sorts of new things," and... well... this is where that led. It has... failed miserably. It wasn't too bad before, but now Alex has apparently just gone out and gotten drunk.
So today a trampoline appeared in our lounge. Noone was really sure where it came from. Some people said Henderson had a trampoline, and we must have stolen it from them, while others said Shorey had a trampoline, and we stole it from them. Eventually it came out that some guy named Noah from Thompson stole it from Shorey and gave it to us, because Shorey was always jumping on it and annoying the Thompson kids below them, so they gave it to us because there's no one below us. Only it turns out, Shorey actually originally stole the trampoline from Henderson so that they could make slam dunks in their basketball hoop.
Neat little problem we got a bit ago in math:
(I should note that we have not talked about Lebesgue measure yet, I assume we won't for quite a while, but we have defined measure 0 for subsets of R.)
So consider the function f(x)=1/denom(x) for x∈Q, 0 otherwise. Only modify it so that f(0)=0, and actually we're going to consider fr (exponentiation, not iteration) for r≥1. As you know, fr is continuous precisely on the irrationals and 0.
Then:
1. Show fr not differentiable on the irrationals for r≤2.
2. For what r is f differentiable at 0?
3. (The neat part!) Show that fr is differentiable almost everywhere for r>2.
(Hint: First show that A⊆R is measure 0 iff you can cover it with a countable collection of open intervals such that each element of A is in infinitely many elements of this cover, and so that the total length of the cover is finite.)
I think I'll start referring to people here using their house in place of their last name when this is unambiguous, e.g. "Lisa of Flint".
-Sniffnoy
