Dan LuuFeb 2, 2024

In other https://danluu.com/discontinuities/ news, I'm surprised to see that so many people round their age off in (many? most?) age surveys, e.g., see this poll on the age of WordPress users, which has spikes at 25, 30, 40, 45, 50, and 55:

https://wordpress.org/news/files/2023/05/2022-Annual-Survey-Final.pdf

Suspicious discontinuities

Show threadJay DaigleFeb 2, 2024
@danluu I'm going to steal a couple examples from that post for my next calc 1 class. (If I'd seen it three weeks ago, I'd have stolen them for this one.)
Show threadDan LuuFeb 3, 2024
@profjaydaigle Great! What are you using the examples to illustrate? Or is it just that they're generically weird/fun?
Show threadJay Daigle

@danluu

I organize the course around the idea of approximation. With limits, we're asking if knowing f(a) gives us a good estimate of f(x) for x~a.

In continuous functions, which include most reasonable algebraic formulas, it does, but in lots of real-world functions it doesn't—I use that same graph of car prices. (I.e. knowing a car has "about" 40k miles isn't enough information, if you don't know whether it's just above or just below.)

Feb 3, 2024 at 2:45amWeb
Show threadJay DaigleFeb 3, 2024

@danluu

But most of the examples we actually study, and pretty much all the computations we do, are looking at removable discontinuities, where there's just one missing or misplaced point. And that idea tends to sound artificial and made-up: "We took this reasonable pattern and then changed a point to mess with you."

But the p-value chart and the crack chart are both perfect examples of removable discontinuities, so I can use them to illustrate that point.

Show threadDan LuuFeb 4, 2024
@profjaydaigle Oh, cool. I do remember thinking that a lot of the counterexamples we'd think about in calculus seeming very disconnected from reality. I didn't mind since I didn't it to be connected to anything to enjoy learning it, but it would've been even more interesting with some kind of connection to "the real world".