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Calculus using proximities: a mathematical approach in which students can actually prove theorems

Richard O’Donovan (2017)

Open Mathematics

Teaching and learning calculus are notoriously difficult and the didactic solutions may involve resorting to intuitive but vague definitions or informal gestures offered as proofs. The teaching literature is rife with examples of metaphors, adverb manipulations and descriptions of what happens “just before” the limit. It is then difficult to leave the domain of the mental image, thus losing the training in rigour. The author (with Karel Hrbacek and Olivier Lessmann) has endeavoured a radically different...

Equation f ( p ( x ) ) = q ( f ( x ) ) for given real functions p , q

Oldřich Kopeček (2012)

Czechoslovak Mathematical Journal

We investigate functional equations f ( p ( x ) ) = q ( f ( x ) ) where p and q are given real functions defined on the set of all real numbers. For these investigations, we can use methods for constructions of homomorphisms of mono-unary algebras. Our considerations will be confined to functions p , q which are strictly increasing and continuous on . In this case, there is a simple characterization for the existence of a solution of the above equation. First, we give such a characterization. Further, we present a construction...

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