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MATHEMATICAL MYSTERIES
The Function That Is Broken Everywhere
How Dirichlet Changed What Mathematicians Thought a Function Could Be
A function sounds like a calm object.
You give it a number. It gives you another number. We usually imagine a curve, perhaps smooth, perhaps sharply bent, perhaps with a few jumps. Even when a function behaves badly, we often expect the bad behavior to occur at special points.
Then comes the Dirichlet function.
It is one of the simplest functions in mathematics, and one of the most disturbing. It does not need a complicated formula. It does not need infinite series, trigonometric tricks, or advanced machinery. It only asks one question:
Is the input rational or irrational?
If the answer is rational, the function gives 1.
If the answer is irrational, the function gives 0.
That is all.
And from this innocent rule comes a function that is discontinuous everywhere, cannot be drawn as an ordinary curve, has no point of local calm, defeats…
