Why did discontinuous Fourier series lead to new kinds of integrals?
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
The investigation of Fourier series revealed that these series could represent discontinuous functions, which challenged the existing mathematical framework that was largely limited to continuous functions.12
As mathematicians encountered these increasingly "exotic" functions, the traditional methods of integration proved insufficient to handle the resulting discontinuities. To address these limitations, new, more robust definitions of the integral were developed to accommodate highly discontinuous functions. This led to the formulation of the Riemann integral in 1854 and, subsequently, the Lebesgue integral in 1902. The Lebesgue integral, in particular, revolutionized integration by utilizing a new theory of measure, which allowed for a much broader scope of functions to be integrated and became instrumental in the further development of Fourier analysis.21