# A single hidden layer feedforward network with only one neuron in the hidden layer can approximate any univariate function

arXiv: https://arxiv.org/abs/1601.00013 · HTML: https://mlsanity.com/paper/1601.00013/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Dec 2015.

## Built on by (most cited first, 6 of 6)

- [Addressing common misinterpretations of KART and UAT in neural network literature](https://mlsanity.com/paper/2408.16389/) (2024)
- [A Unified and Constructive Framework for the Universality of Neural Networks](https://mlsanity.com/paper/2112.14877/) (2021)
- [On Explicit Super-Expressive Approximation for Neural Networks](https://mlsanity.com/paper/2607.06781/) (2026)
- [Learnability of high-dimensional targets by two-parameter models and gradient flow](https://mlsanity.com/paper/2402.17089/) (2024)
- [Universal Approximation Theorem for Input-Connected Multilayer Perceptrons](https://mlsanity.com/paper/2601.14026/) (2026)
- [Arbitrary-Accuracy Neural Approximation with Optimal Neuron Count and Near-Optimal Bit Complexity](https://mlsanity.com/paper/2609.35628/) (2026)

## Abstract (arXiv)

The possibility of approximating a continuous function on a compact subset of the real line by a feedforward single hidden layer neural network with a sigmoidal activation function has been studied in many papers. Such networks can approximate an arbitrary continuous function provided that an unlimited number of neurons in a hidden layer is permitted. In this paper, we consider constructive approximation on any finite interval of ℝ by neural networks with only one neuron in the hidden layer. We construct algorithmically a smooth, sigmoidal, almost monotone activation function σ providing approximation to an arbitrary continuous function within any degree of accuracy. This algorithm is implemented in a computer program, which computes the value of σ at any reasonable point of the real axis.