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The uniform continuity theorem (UCT) states that every pointwise continuousreal-valued function on the unit interval is uniformly continuous. Inconstructive mathematics, UCT is stronger than the decidable fan theorem (DFT);however, Loeb [Ann. Pure Appl. Logic, 132(1):51-66, 2005] has shown that thetwo principles become equivalent with a suitable coding of "continuousfunctions" as type-one objects. The question remains whether DFT can becharacterised by a weaker version of UCT using a natural subclass of pointwisecontinuous functions without such a coding. We show that when "pointwisecontinuous" is replaced with "having a continuous modulus", UCT becomesequivalent to DFT. We also show that this weakening of UCT is equivalent to asimilar principle for real-valued functions on the Cantor space$\{0,1\}^{\mathbb{N}}$. These results extend Berger's characterisation of DFTby the similar principle for functions from $\{0,1\}^{\mathbb{N}}$ to$\mathbb{N}$, and unifies these characterisations of DFT in terms of functionshaving continuous moduli. Furthermore, we directly show that the continuousreal-valued functions on the unit interval having continuous moduli are exactlythose functions which admit the coding of "continuous functions" due to Loeb.Our result allows us to interpret her work in the usual context of mathematics.

Decidable fan theorem and uniform continuity theorem with continuous moduli