Thomae's Function: The Wobbly Number Line!

Thomae's function, formally defined as f(x) = 1/q for x = p/q (where p is an integer, q is a positive integer, and p and q are coprime) and f(x) = 0 for irrational x, presents a unique landscape on the real number line. Unlike simpler functions, its graph is not a smooth curve but a series of discrete points at rational numbers, each with a height determined by the denominator of its reduced fraction, set against a background of absolute zero for all irrational numbers.

This structure is a direct consequence of the definition of rational numbers and the property that every rational number has a unique representation in lowest terms. The function's value at a rational number x = p/q is precisely 1/q. For instance, f(1/2) = 1/2, f(1/3) = 1/3, f(2/3) = 1/3, f(1/4) = 1/4, and f(3/4) = 1/4.

This means that as the denominator q increases, the height of the 'peak' at p/q decreases, approaching zero. The function is thus bounded between 0 and 1.

Carl Thomae introduced this function in the late 19th century, not as an abstract curiosity, but as a powerful pedagogical tool. He used it to illustrate a critical point in Riemann integration theory: that a function could possess an infinite number of discontinuities and still be integrable. This was a significant departure from earlier intuitions that suggested integrability implied a certain 'smoothness' or limited number of breaks.

Thomae's function is, in fact, continuous at every irrational number and discontinuous at every rational number. This is because for any irrational number, you can find a sequence of rational numbers that converge to it, and the function values at these rationals (1/q) converge to 0, matching the function's value at the irrational number. Conversely, for any rational number, no matter how close you get with other rational numbers, the function values will jump to 0 at the irrationals, preventing convergence to a single value.

This function is a modification of the Dirichlet function, which is 1 for all rationals and 0 for all irrationals, making Thomae's function a more nuanced exploration of number properties.

The importance of Thomae's function lies in its elegant demonstration of the properties of the real number line. It vividly illustrates the concept of density: both rational and irrational numbers are dense in the real numbers, meaning between any two distinct real numbers, there exists both a rational and an irrational number. Thomae's function shows that while rational numbers are countable (meaning they can be put into a one-to-one correspondence with the natural numbers), they are 'sparse' in a measure-theoretic sense compared to the uncountable set of irrational numbers.

The set of rational numbers has Lebesgue measure zero, which is why the integral of Thomae's function over any interval is zero, despite the function having non-zero values at infinitely many points. This function is a cornerstone in real analysis, helping students grasp concepts like continuity, discontinuity, limits, and integration in a rigorous manner, and it has applications in areas like Fourier analysis and signal processing where understanding discontinuous functions is vital.

The nicknames 'popcorn function' and 'raindrop function' are apt metaphors for Thomae's function. The 'popcorn' analogy captures the idea of discrete, upward bursts at rational numbers, with the height of the pop (1/q) diminishing as the denominator gets larger, much like popcorn kernels popping at different heights. The 'raindrop' analogy highlights the vast, flat expanse of zero where irrational numbers reside, punctuated by tiny, scattered 'drops' (the rational numbers) that are almost lost in the overall emptiness.

John Horton Conway's name, 'Stars over Babylon,' evokes a similar image of a vast, dark expanse (Babylon) dotted with numerous, small points of light (stars), representing the irrationals and rationals, respectively. These names help visualize the function's peculiar graph and its distribution of values across the number line.

Thomae's function is closely related to other fundamental concepts in mathematics. The Dirichlet function, as mentioned, is its simpler predecessor. The concept of 'coprime' or 'relatively prime' is crucial for ensuring the function is well-defined, as each rational number has a unique representation in its lowest terms.

The function's behavior at rational and irrational numbers directly relates to the topological properties of the real numbers, specifically the notions of open and closed sets, and the difference between countable and uncountable sets. Its integrability, despite infinite discontinuities, is a key topic in measure theory and Lebesgue integration, which generalize Riemann integration. Understanding Thomae's function provides a solid foundation for exploring more complex functions and their properties in advanced calculus and analysis.