## Web ads

I apologize to anyone who went to my website recently and found a bunch of annoying ads. I figured it was the product of either a) the hosting service pulling crap on me and saying “HAHAHA, THIS IS WHY IT’S FREE!” or b) h4x0rs. Turns out it’s the hosting service, but they only do it if you haven’t managed your website in a while. Shows you how much time I dedicate to it. In any case, it should be fine now, so take a look!

Proof I came up with some weeks ago. If a ruler covers the entire line and has marks at the dyadic rationals (multiples of a power of 2) that are larger if the power of 2 involved is higher, then any open interval has exactly one mark bigger than all of the others.

So for those of you who didn’t see the big link above this text, I got a channel on twitch.tv! I’ll mostly be using it to stream The Binding of Isaac for now, but I’ll most likely be playing some other games on there as well, so check it out!

## Sequential Discontinuity

I just proved a couple of interesting theorems relating to a concept I’ve defined and call the *sequential discontinuity.* Anyone who has studied calculus knows of three or four types of discontinuity. The first is the removable discontinuity, where a function is undefined or “wrongly” defined at a single point. The second is the jump discontinuity, where a function approaches a different value from either side of the discontinuity. The third type is the essential discontinuity, which is basically anything else, so teachers tend to further subdivide it into vertical asymptotes (which make up the vast majority of essential discontinuities) and oscillating discontinuities such as the discontinuity of sin(1/x) at x = 0 (even if the function is altered such that f(0) = 0). However, not all essential discontinuities fall into this category.

Consider the following function: f(x) = 0 for all x such that log(x) is not an integer and is undefined otherwise. (This is the base 10 logarithm). This function has obvious removable discontinuities at x = 10^k for integer k, but it also has a discontinuity at x = 0. This is because the formal definition of the limit requires an open interval of x-values around 0 where all f(x) except possibly f(0) are undefined, and no such interval exists. In layman’s terms, you can’t put a mark anywhere past 0 on the number line without a power of ten between 0 and the mark, no matter how close it is to 0. This discontinuity is not an asymptote and does not involve oscillation (the value of f(x) never changes where it exists).

A sequential discontinuity exists in two cases: at the limit of a convergent sequence of x-values where the function is undefined, and at the limit of a convergent sequence of x-values where the sequence of the corresponding function values is unbounded. Note that the second case especially can be a different kind of discontinuity as well; for instance, the discontinuity at x = 0 for f(x) = 1/x is sequential, as we can define the sequence x = 1, 1/2, 1/3, …, which converges to 0 and has function values that increase without bound. It is also a vertical asymptote, a simpler case.

The interesting thing about sequential discontinuities is that sometimes they are “removable” in the sense that the function containing them could be made continuous by changing countably many values (strangely not including the value at the discontinuity). However, they have undefined limits and as such are *not* removable.

I have here a link to a PDF proof and more formal definition. One small lemma is required for the second case but is not proven, so I will prove it here. An unbounded sequence has for every real number k and natural number M a value m > M such that a_m > k. To prove this, choose i such that a_i > k. If a_i > M, then m = i. If a_i <= M, then let K be the maximum of a_1, a_2, …, a_M. We are guaranteed K >= a_i > k. Then there must be a j such that a_j > K, and j must be greater than M (otherwise K would not be the maximum), so let m = j and a_m > k.

https://www.dropbox.com/s/rsc9q0o0ta1zyd3/Theorem—SequentialDiscontinuity.pdf

So I decided yesterday to get my own website! I’ll post the finished versions of all of my games there, as well as more info about myself, etc. Check it out!

## Stoichios 1.0 Out!

Yes, another game release within the span of a few weeks. As it turns out, this is a game I started in 2011. It was up to release candidate 3, but then I lost access to Windows computers in general. No worries; I’ve been able to work out the last few bugs and get the game out.

The game is a card game based on the classical/Greek elements (earth, air, fire, and water) and otherwise plays similarly to Mau Mau or Uno. Single-player and multiplayer modes are included (both local and online). There is also AI. Overall, it’s a pretty fun game, and it has what I consider some of my best music in it as well.

The game is a standalone .exe in a .zip archive; make sure to unzip it first in order to get the DLL files out. The soundtrack .mp3s are included if you want it in iTunes. Let me know if anything goes wrong.

Download: https://www.dropbox.com/s/5m3u90n7e38lu8k/Stoichios_100.zip

## Resonance 1.0 Out!

What you see above is both the logo of my newest game, called Resonance, and every possible screenshot from within the game. If it doesn’t look like it’s loaded yet, it probably has. The game takes place in complete darkness.

Resonance is a game about solving puzzles without the luxury of sight. The game takes place in a maze on a standard tile grid (tiles are around 4’ wide in my mind). Instead of being able to see the maze around you, you can hear what’s in front of you, represented as pitches relative to a base of 440 Hz. Distance corresponds to the pitch and volume of each sound, and object type corresponds to the timbre.

The game has 50 levels ranging from pretty difficult to infuriatingly difficult. Headphones and graph paper are both recommended.

In theory, a blind person could play this game without being hindered (as long as he or she could find the control keys). A deaf person, on the other hand, would know absolutely nothing about the state of the game.

Download links:

- Mac: http://www.mediafire.com/?p3eqgl30i8ibpg2
- Windows: http://www.mediafire.com/?0w590pyn1pc0p7e
- Linux: http://www.mediafire.com/?vrzu22vqhr9ham3
- Source code: http://www.mediafire.com/?dp9yym9k3738s5m
- Soundtrack (MP3): http://www.mediafire.com/?kkfh74csyx0bo0i

## YouTube

I’m now on YouTube! Find me atÂ http://www.youtube.com/user/Whirligig231. I’m not entirely sure exactly what will go up on here, but it’ll be worth a look.