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Level sets of continuous functions increasing with respect to each variable

100%

Sajbura K.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2005

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tom 25

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nr 1

19-26

EN

We are going to prove that level sets of continuous functions increasing with respect to each variable are arcwise connected (Theorem 3) and characterize those of them which are arcs (Theorem 2). In [3], we will apply the second result to the classical linear functional equation φ∘f = gφ + h (cf., for instance, [1] and [2]) in a case not studied yet, where f is given as a pair of means, that is so-called mean-type mapping.

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The Basic Existence Theorem of Riemann-Stieltjes Integral

100%

Nakasho K., Narita K., Shidama Y.

Formalized Mathematics

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2016

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tom 24

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nr 4

253-259

EN

In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15], [12], [10], and [11].

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Functions characterized by images of sets

51%

Ciesielski K., Dikrajan D., Watson S.

Colloquium Mathematicum

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1998

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tom 77

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nr 2

211-232

EN

For non-empty topological spaces X and Y and arbitrary families $\cal A$ ⊆ $\cal P(X)$ and $\cal B ⊆ \cal P(Y)$ we put $\cal C_{\cal A,\cal B}$={f ∈ $Y^X$ : (∀ A ∈ $\cal A$)(f[A] ∈ $\cal B)$}. We examine which classes of functions $\cal F$ ⊆ $Y^X$ can be represented as $\cal C_{\cal A,\cal B}$. We are mainly interested in the case when $\cal F=\cal C(X,Y)$ is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class $\cal F=\cal C$(X,ℝ) is not equal to $\cal C_{\cal A,\cal B}$ for any $\cal A$ ⊆ $\cal P(X)$ and $\cal B$ ⊆ $\cal P$(ℝ). Thus, $\cal C$(X,ℝ) cannot be characterized by images of sets. We also show that none of the following classes of real functions can be represented as $\cal C_{\cal A,\cal B}$: upper (lower) semicontinuous functions, derivatives, approximately continuous functions, Baire class 1 functions, Borel functions, and measurable functions.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Formalized Mathematics

1 Ciesielski K.

1 Dikrajan D.

1 Nakasho K.

1 Narita K.

1 Sajbura K.

1 Shidama Y.

1 Watson S.

1 2016

1 2005

1 1998

Level sets of continuous functions increasing with respect to each variable

The Basic Existence Theorem of Riemann-Stieltjes Integral

Functions characterized by images of sets