Student Example

Graphic Organizer Leonhard Euler and Jean d’Alembert: Argue over solutions 1 𝐹(𝑥, 𝑡) = [𝑓(𝑥 + 𝑡) + 𝑓(𝑥 − 𝑡)] 2

D’Alembert and Euler: f must be continuous or discontinuious

Daniel Bernouslli, Euler, d’Alemvert, Joseph Lagrange: Solutions must have a specific form but is rejected. 𝜋𝑥 2𝜋𝑥 𝑓(𝑥) = 𝑎1 ( ) + 𝑎2 ( )+⋯ 𝐿 𝐿

Joseph Fourier: A bounded function has the form ∞ 1 𝑓(𝑥) = 𝑎0 + ∑ 𝑎𝑛 cos 𝑛𝜋𝑥 /𝑎 + 𝑏𝑛 sin 𝑛𝜋𝑥 /𝑎 2 𝑛=1

Joseph Fourier: A function is continuous the same way it is today

Fourier: Arbitrary functions behave well ∞ 1 𝑏 𝑓(𝑥) = ∫ 𝑓(𝑎)𝑑𝑎 ∫ cos(𝑝𝑥 − 𝑝𝑎) 𝑑𝑝 2𝜋 𝑎 −∞

Fourier: A general function is smooth

Fourier, Lagrange, Henri Lebesgue: Coefficents attempted to be validated by Fourier. Lagrange improved this. Henri gave a proper definition.

Integrals change from area to antiderivative.

Augustin Cauchy: Provided world modern definition of continuity and defined definite integral.

Cauchy: Defines Cauchy sum as definite integral.

Cauchy: Functions are equations.

Discontinuous functions have the form 𝑛

𝑓(𝑥) = ∑ 𝑋𝐼𝑟 (𝑥)𝑔𝑟 (𝑥)𝑟 𝑟=1

Peter Gustav Lejeune-Dirichlet: Gave first proof of convergence.

Lejeune-Dirichlet: Believed proof would work for discontinuities

Rudolf Lipschitz: Tried to improve Dirichlet’s proof.

Dirichlet and George Cantor: Proof extends to sets created by Cantor

Dirichlet: Introduced function 1 𝑥 𝑖𝑠 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑓(𝑥) = { 0 𝑥 𝑖𝑠 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙