The uniform continuity theorem $(\mathsf{UCT})$ states that every pointwise continuous real-valued function on the unit interval is uniformly continuous. In constructive mathematics, $\mathsf{UCT}$ is strictly stronger than the decidable fan theorem $(\mathsf{DFT})$, but Loeb [17] has shown that the two principles become equivalent by encoding continuous real-valued functions as type-one functions. However, the precise relation between such type-one functions and continuous real-valued functions (usually described as type-two objects) has been unknown. In this paper, we introduce an appropriate notion of continuity for a modulus of a continuous real-valued function on [0, 1], and show that real-valued functions with continuous moduli are exactly those functions induced by Loeb's codes. Our characterisation relies on two assumptions: (1) real numbers are represented by regular sequences (equivalently Cauchy sequences with explicitly given moduli); (2) the continuity of a modulus is defined with respect to the product metric on the regular sequences inherited from the Baire space. Our result implies that $\mathsf{DFT}$ is equivalent to the statement that every pointwise continuous real-valued function on [0, 1] with a continuous modulus is uniformly continuous. We also show that $\mathsf{DFT}$ is equivalent to a similar principle for real-valued functions on the Cantor space ${\{0,1\}}^{\mathbb{N}}$. These results extend Berger's [2] characterisation of $\mathsf{DFT}$ for integer-valued functions on ${\{0,1\}}^{\mathbb{N}}$ and unify some characterisations of $\mathsf{DFT}$ in terms of functions having continuous moduli.

中文翻译：

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可判定扇形定理和具有连续模的一致连续性定理

一致连续性定理 $(\mathsf{统一通信技术})$指出单位区间上的每个逐点连续实值函数是一致连续的。在构造性数学中，$\mathsf{统一通信技术}$ 严格强于可判定扇形定理 $(\mathsf{离散傅立叶变换})$，但 Loeb [17] 已经表明，通过将连续实值函数编码为第一类函数，这两个原则变得等效。然而，此类第一类函数与连续实值函数（通常被描述为第二类对象）之间的确切关系尚不清楚。在本文中，我们为 [0, 1] 上的连续实值函数的模引入了适当的连续性概念，并表明具有连续模的实值函数正是由 Loeb 代码导出的那些函数。我们的表征依赖于两个假设：（1）实数由规则序列（等效于具有明确给定模数的柯西序列）表示；(2) 模数的连续性是根据从 Baire 空间继承的正则序列上的乘积度量来定义的。我们的结果意味着$\mathsf{离散傅立叶变换}$等价于具有连续模数的 [0, 1] 上的每个逐点连续实值函数均匀连续的陈述。我们还表明$\mathsf{离散傅立叶变换}$ 等价于康托空间上实值函数的类似原理 ${\{0,1\}}^{\mathbb{N}}$. 这些结果扩展了伯杰的 [2] 表征$\mathsf{离散傅立叶变换}$ 对于整数值函数 ${\{0,1\}}^{\mathbb{N}}$ 并统一一些特征 $\mathsf{离散傅立叶变换}$ 就具有连续模量的函数而言。