caa a22 4500 605461325 CHVBK 20210128100242.0 cr unu---uuuuu 210128e20150701xx s 000 0 eng 10.1007/s10114-015-3759-x doi (NATIONALLICENCE)springer-10.1007/s10114-015-3759-x On a problem of Hornik [Elektronische Daten] [Ting Xie, Fei Cao] In 1991, Hornik proved that the collection of single hidden layer feedforward neural networks (SLFNs) with continuous, bounded, and non-constant activation function σ is dense in C(K) where K is a compact set in ℝ s (see Neural Networks, 4(2), 251-257 (1991)). Meanwhile, he pointed out "Whether or not the continuity assumption can entirely be dropped is still an open quite challenging problem”. This paper replies in the affirmative to the problem and proves that for bounded and continuous almost everywhere (a.e.) activation function σ on ℝ, the collection of SLFNs is dense in C(K) if and only if σ is un-constant a.e.. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg, 2015 Neural networks nationallicence approximation nationallicence activation function nationallicence Xie Ting Department of Mathematics and Information Sciences, China Jiliang University, 310018, Hangzhou, P. R. China aut Cao Fei Department of Mathematics and Information Sciences, China Jiliang University, 310018, Hangzhou, P. R. China aut Acta Mathematica Sinica, English Series Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 31/7(2015-07-01), 1141-1148 1439-8516 31:7<1141 2015 31 10114 https://doi.org/10.1007/s10114-015-3759-x text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10114-015-3759-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Xie Ting Department of Mathematics and Information Sciences, China Jiliang University, 310018, Hangzhou, P. R. China aut NATIONALLICENCE 700 1- Cao Fei Department of Mathematics and Information Sciences, China Jiliang University, 310018, Hangzhou, P. R. China aut NATIONALLICENCE 773 0- Acta Mathematica Sinica, English Series Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 31/7(2015-07-01), 1141-1148 1439-8516 31:7<1141 2015 31 10114