We consider X as an uncountable compact set within [0, 1] and Y as a compact interval in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb R$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>. It has been shown that, for a typical continuous function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Psi :X\longrightarrow Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ψ</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">⟶</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> and under specific additional assumptions, the generalized packing dimension of its graph equals the sum of the generalized packing dimensions of X and Y. Moreover, we explore the decomposition of continuous functions on [0, 1] based on the upper box dimension and the generalized packing dimension. Furthermore, we present results regarding the generalized packing dimension of graphs arising from the sum and product of continuous functions. Our main argument establishes that, for a given real number <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> between the generalized packing dimensions of [0, 1] and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$[0,1]\times [0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and with specific conditions, there exists a real-valued function in a set of continuous functions on [0, 1] that can be decomposed into the sum and product of two continuous real-valued functions, with the generalized packing dimension of each function’s graph being <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>.

The role of generalized packing and box dimensions in the study of continuous functions

期刊论文
发表日期：

2025年11月21日

查看全文

Rim Achour,
Bilel Selmi,
Zhiming Li,
Binyan Yu

摘要

We consider X as an uncountable compact set within [0, 1] and Y as a compact interval in $\mathbb R$ $R$ . It has been shown that, for a typical continuous function $\Psi :X\longrightarrow Y$ $Ψ : X ⟶ Y$ and under specific additional assumptions, the generalized packing dimension of its graph equals the sum of the generalized packing dimensions of X and Y. Moreover, we explore the decomposition of continuous functions on [0, 1] based on the upper box dimension and the generalized packing dimension. Furthermore, we present results regarding the generalized packing dimension of graphs arising from the sum and product of continuous functions. Our main argument establishes that, for a given real number $\gamma $ $γ$ between the generalized packing dimensions of [0, 1] and $[0,1]\times [0,1]$ $[0, 1] \times [0, 1]$ and with specific conditions, there exists a real-valued function in a set of continuous functions on [0, 1] that can be decomposed into the sum and product of two continuous real-valued functions, with the generalized packing dimension of each function’s graph being $\gamma $ $γ$ .