Eduard Heine is not among the best-known mathematicians, even though his name is attached to a theorem on the uniform continuity of continuous functions on a closed interval.
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Until the early 19th century, everything seemed continuous. Trigonometric series brought discontinuous functions onto the scene. They prompted Riemann to generalize the notion of the integral, led Weierstrass to clarify the notion of continuity, and spurred Darboux to study derivatives.
The Cauchy–Schwarz inequality takes many forms: arithmetic, integral and geometric; it even appears in probability theory. Let us trace its development, from Cauchy's numerical formulation around 1820 to its general form a century later.
From Cauchy to Weierstrass, rigor steadily took hold in analysis throughout the 19th century. A variety of counterexamples swept away mistaken beliefs and forced mathematicians to define the relevant concepts more precisely. Some "monstrous" functions were introduced, fascinating to some and repellent to others.
Repeating a random experiment with two possible outcomes a large number of times leads to probabilities that are difficult to predict directly, let alone calculate explicitly. Yet the de Moivre–Laplace theorem provides an excellent approximation.
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