Continuous but not differentiable

Today I learned about functions that are continuous but not differentiable. I'm mostly sharing this because it came up in the context of a hilarious why, math, why would you do this to us kind of conversation and I figured many people in CSC236 could relate to that feel. I don't think it has anything to do with anything in CSC236 other than that, to be honest, but it's kind of neat!




Keep in mind that continuity on a curve can basically be defined as looking at two sides of a given point on a curve. If the function is defined at that point, and the thing that it's defined as is also equal to the left and right limits (which have to exist), then it's continuous. It's like a chain; if the link you want is closed, and it links up properly to the links on either side, then the chain is connected; otherwise, it's not.

A function is differentiable at a point x_0 if it's continuous there. In particular a function that is differentiable must be continuous for all points in the domain. (You can see how it could be easy to get around that; just redefine the domain to exclude all the points where it isn't continuous and therefore not differentiable.) However, the converse is not necessarily true: while a differentiable function must be continuous, just because a function is continuous does not directly imply that it's differentiable.

Suppose you take an equilateral triangle. It's continuous - the straight edges are obviously continuous, and the vertices, which are basically cusps, are continuous (the points exist on the curve, and their left and right limits exist, and also equal the value of the cusp-point). It certainly seems differentiable on every part of it except for the vertices, because those are cusps.

Suppose now you take every one of those sides and divide them into three. Use the middle segment as the base for a new, smaller equilateral triangle, and then later delete that middle segment.

Again, it's continuous everywhere, and differentiable everywhere except the vertices. And then you just keep doing that for awhile. A long while.

But every time you do, the parts where it is differentiable decrease and decrease because you keep adding more and more triangles.

Anyway none of this is novel or anything, I just think it's really cool. And also pretty snowflake.

tl;dr: Weierstrass is a troll.