A continuous function is one where a small variation of its argument induces at most a small variation of its value, so that the function has no abrupt jumps or breaks. Bernard Bolzano gave a form of the epsilon-delta definition of continuity in 1817. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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Source Augustin-Louis Cauchy (Wikipedia)
Source Continuous Function (Wikipedia)
The theorem's hypothesis is that the function is continuous on the interval, so its subject is the continuity concept itself.
Additional Source Intermediate Value Theorem (Wikipedia)introductory paragraph
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Source Continuous Function (Wikipedia)
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Source Continuous Function (Wikipedia)
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1. Continuous Function (Wikipedia)
Wikimedia Foundationlead paragraph
a small variation of its argument induces at most a small variation of its value. This implies there are no abrupt changes in value, known as discontinuities.
History section
A form of the epsilon-delta definition of continuity was first given by Bernard Bolzano in 1817.
Attributed To: Bernard Bolzano, Lead
A form of the epsilon-delta definition of continuity was first given by Bernard Bolzano in 1817.
View the Source Augustin-Louis Cauchy (Wikipedia)
Wikimedia FoundationAssociated With: Augustin-Louis Cauchy, lead paragraphQuote, Associated With: Augustin-Louis Cauchy, lead paragraph
He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real analysis)
Intermediate Value Theorem (Wikipedia)
Wikimedia FoundationAssociated With: Intermediate Value Theorem, introductory paragraphQuote, Associated With: Intermediate Value Theorem, introductory paragraph
if f is a continuous function whose domain contains the interval [a, b] and s is a number such that f(a)<s<f(b), then there exists some x between a and b such that f(x)=s
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