Elementary proof of Funahashi's theorem

Funahashi established that the space of two-layer feedforward neural networks is dense in the space of all continuous functions defined over compact sets in $n$-dimensional Euclidean space. The purpose of this short survey is to reexamine the proof of Theorem 1 in Funahashi \cite{Funahashi}. The Tietze extension theorem, whose proof is contained in the appendix, will be used. This paper is based on harmonic analysis, real analysis, and Fourier analysis. However, the audience in this paper is supposed to be researchers who do not specialize in these fields of mathematics. Some fundamental facts that are used in this paper without proofs will be collected after we present some notation in this paper.