| Paul Bois-Reymond|
| Emil du Bois-Reymond|
| April 7, 1889, Freiburg im Breisgau, Germany|Paul du Bois-Reymond Wikipedia
Paul David Gustav du Bois-Reymond (2 December 1831 – 7 April 1889) was a German mathematician who was born in Berlin and died in Freiburg. He was the brother of Emil du Bois-Reymond.
His thesis was concerned with the mechanical equilibrium of fluids. He worked on the theory of functions and in mathematical physics. His interests included Sturm–Liouville theory, integral equations, variational calculus, and Fourier series. In this latter field, he was able in 1873 to construct a continuous function whose Fourier series is not convergent. His lemma defines a sufficient condition to guarantee that a function vanishes almost everywhere.
Du Bois-Reymond also established that a trigonometric series that converges to a continuous function at every point is the Fourier series of this function. He also discovered a proof method that later became known as the Cantor's diagonal argument. His name is also associated with the fundamental lemma of calculus of variations of which he proved a refined version based on that of Lagrange.
Paul du Bois-Reymond developed a theory of infinitesimals:Théorie générale des fonctions (Nice : Impr. niçoise, 1887) (translated in French from the original German by G. Millaud and A. Girot)
De Aequilibrio Fluidorum (PhD Thesis, 1859)