Tag Archives: monsters of real analysis

TWSB: Continuous, Continuous Everywhere but Not a Derivative to Find

WOAH, I’m pretty sure I’ve heard of this before but didn’t know what it was.


This is a Weierstrass function. Pretty, isn’t it?

Know what else is cool about it? While it’s continuous everywhere, it’s differentiable nowhere.

This challenged the notion that every continuous function was differentiable except on a small set of isolated points when it was discovered and published by Karl Weierstrass in 1872 (though some suggest that Riemann had made this discovery in 1871).

Haha, and I didn’t know this, but examples like this (functions that are continuous but not differentiable except only on a set of points of measure zero) are called monsters of real analysis.

That would be a FANTASTIC thrash metal band name. They could tour with Step Reckoner (the other fictional metal band I came up with, named after Leibniz’ calculating machine).