Abstract
Everybody has an intuitive concept of continuity of a process or of a function.
Now we shall make the concept of continuity of a function precise, and we shall describe the main general properties of continuous functions.
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- 1.
We recall that \(U_{E}(a)=E\cap U(a)\).
- 2.
If \(a\) is a discontinuity, then \(a\) must be a limit point of the set \(E\). It may happen, however, that all the points of \(E\) in some neighborhood of \(a\) lie on one side of \(a\). In that case, only one of the limits in this definition is considered.
- 3.
P.G. Dirichlet (1805–1859) – great German mathematician, an analyst who occupied the post of professor ordinarius at Göttingen University after the death of Gauss in 1855.
- 4.
B.F. Riemann (1826–1866) – outstanding German mathematician whose ground-breaking works laid the foundations of whole areas of modern geometry and analysis.
- 5.
We recall that \(C(E)\) denotes the set of all continuous functions on the set \(E\). In the case \(E=[a, b]\) we often write, more briefly, \(C[a, b]\) instead of \(C ([a, b] )\).
- 6.
Here \(f(a)\leq f(b)\) if \(f\) is nondecreasing, and \(f(b)\leq f(a)\) if \(f\) is nonincreasing.
- 7.
For this reason the modulus of continuity is usually considered for \(\delta\geq 0\), setting \(\omega(0)=\omega(+0)\).
- 8.
Ch.J. de la Vallée Poussin (1866–1962) – Belgian mathematician and specialist in theoretical mechanics.
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© 2015 Springer-Verlag Berlin Heidelberg
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Zorich, V.A. (2015). Continuous Functions. In: Mathematical Analysis I. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48792-1_4
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DOI: https://doi.org/10.1007/978-3-662-48792-1_4
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