Short answers, pulled from the story.
Zeno of Elea was the Greek philosopher from the 5th century BCE who argued that motion was impossible by claiming one must travel half the distance, then half of the remaining distance, and so on forever. This paradox implicitly contained the concept of an infinite geometric sum that would eventually evolve into the rigorous study of limits.
The formal birth of mathematical analysis occurred in the 17th century when Isaac Newton and Gottfried Wilhelm Leibniz independently developed infinitesimal calculus. Newton worked in England focusing on physical applications while Leibniz worked in Germany developing the notation and algebraic framework that remains standard today.
Bernard Bolzano introduced the modern definition of continuity in 1816 but his work remained obscure until the 1870s. Augustin-Louis Cauchy began to put calculus on a firm logical foundation in 1821 by rejecting loose algebraic assumptions and formulating calculus in terms of geometric ideas and infinitesimals.
Stefan Banach created the theory of normed vector spaces in the 1920s to provide a framework for studying linear operators acting on these spaces. This theory emerged from the need to solve integral equations and understand transformations such as the Fourier transform.
Joseph Fourier developed the theory in the early 19th century that any function could be represented as a sum of sine and cosine waves, a concept now known as the Fourier series. This insight revolutionized the study of partial differential equations and became the backbone of signal processing.
Henri Lebesgue introduced the Lebesgue measure to assign the conventional size of Euclidean geometry to suitable subsets of space. This theory solved problems that Riemann integration could not by requiring the definition of measurable subsets to form a sigma-algebra.