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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The topology of the real line underpins the concept of a limit and hence, among other things, those of continuity and differentiability. It deals with notions of proximity between points that arise from the order relation.
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.
Rolle's theorem, like the celebrated mean value theorem, has been dropped from the secondary-school curriculum. Yet these two major results in analysis are of far more than historical importance. They can be used to solve a great many mathematical problems!
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.
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