Uniform continuity of a product of real functions
emphatic uniform continuity of a product of a function pair
This paper deals with the uniform continuity of a pointwise product of two uniformly continuous real-valued functions defined on a metric space. Let \((X,d)\) be a metric space, for \(\{f,g\}\subseteq \mathbb R^X\) and \(\delta >0\) put NEWLINE\[NEWLINE\lambda(f,g,\delta):=\sup\{|(f(x)-f(p))(g(x)-g(p))| : \{x,p\}\subseteq X\;\text{and}\;d(x,p)<\epsilon\},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\Delta:=\{(f,g)\in \mathbb R^X\times\mathbb R^X : \lim_{n\rightarrow \infty} \lambda(f,g,\frac{1}{n})=0\}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\omega_n(f,x):=\lim_{n\rightarrow\infty} \mathrm{diam} f(B_d(x,\frac{1}{n})).NEWLINE\]NEWLINE The authors say the pair \((f,g)\) has an \textit{emphatically uniformly continuous product} provided that \(\forall \epsilon >0,\;\exists \delta >0\;\text{such that}\;\forall x\in X,\;\forall p\in X,\;d(x,p)<\epsilon\) implies NEWLINE\[NEWLINE\left|\frac{1}{2}(f(x)g(p)+f(p)g(x))-f(x)g(x)\right|<\epsilon.NEWLINE\]NEWLINE Among others, the following results are proved: (1) Let \((X,d)\) be a metric space and let \(\{f,g\}\subseteq \mathbb R^X\), then \((f,g)\) has an emphatically uniformly continuous product if and only if \(fg\) is uniformly continuous and \((f,g)\in\Delta\); (2) Let \((X,d)\) be a metric space and let \(f\) and \(g\) be uniformly continuous real functions on \(X\), them \((f,g)\) has an emphatically uniformly continuous product if and only if \(fg\) is uniformly continuous; and (3) Let \((X,d)\) be a metric space and let \(\{f,g\}\subseteq \mathbb R^X\), then \((f,g)\in \Delta\) if and only if \(\lim_{n\rightarrow\infty} \sup_{x\in X} \omega_n(f,x)\omega_n(g,x)=0.\) The authors also present several examples that help to place properly the results accomplished in the paper.
scientific article; zbMATH DE number 6038699