Properties of continuous functions on a compact
DOI:
https://doi.org/10.15587/1729-4061.2013.16341Keywords:
continuous functions, compactness, The Weierstrass theorem, The Cauchy theoremAbstract
The work deals with the definition of a continuous function, definitions of a continuous function by Cauchy, Heine, in the increment language.
The properties of continuous functions on a compact (on an interval) have been studied comprehensively. The 1st and the 2nd Weierstrass theorems, the 1st and the 2nd Cauchy theorems are presented, as well as the main corollaries of them.
The proofs of theorems and corollaries are presented step by step. Sequentially compact sets are important because continuous functions defined on sequentially compact sets have some very useful properties, which they do not have in general when defined on non-compact sets
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