How To Wreck Your Idea About What ‘Continuous’ Means

This attractive little tweet came across my feed yesterday:

The Popcorn function is continuous at all irrational numbers and discontinuous at all rational numbers. pic.twitter.com/uXGMjdYL5J

— Fermat's Library (@fermatslibrary) October 30, 2017

This function — I guess it’s the “popcorn” function — is a challenge to our ideas about what a “continuous” function is. I’ve mentioned “continuous” functions before and said something like they’re functions you could draw without lifting your pen from the paper. That’s the colloquial, and the intuitive, idea of what they mean. And that’s all right for ordinary uses.

But the best definition mathematicians have thought of for a “continuous function” has some quirks. And here’s one of them. Define a function named ‘f’. Its domain is the real numbers. Its range is the real numbers. And the rule matching things in the domain to things in the range is, as pictured:

• If ‘x’ is zero then
• If ‘x’ is an irrational number then
• If ‘x’ is a rational number, then it’s equal in lowest terms to the whole number ‘p’ divided by the positive whole number ‘q’. And for this ‘x’, then

And as the tweet from Fermat’s Library says, this is a function that’s continuous on all the irrational numbers. It’s not continuous on any rational numbers. This seems like a prank. But it’s a common approach to finding intuition-testing ideas about continuity. Setting different rules for rational and irrational numbers works well for making these strange functions. And thinking of rational numbers as their representation in lowest terms is also common. (Writing it as ‘p divided by q’ suggests that ‘p’ and ‘q’ are going to be prime, but, no! Think of or of .) If you stare at the plot you can maybe convince yourself that “continuous on the irrational numbers” makes sense here. That heavy line of dots at the bottom looks like it’s approaching a continuous blur, at least.

It can get weirder. It’s possible to create a function that’s continuous at only a single point of all the real numbers. This is why Real Analysis is such a good subject to crash hard against. But we accept weird conclusions like this because the alternative is to give up as “continuous” functions that we just know have to be continuous. Mathematical definitions are things we make for our use.