  The uniform continuity theorem (UCT) states that every pointwise continuous
real-valued function on the unit interval is uniformly continuous. In
constructive mathematics, UCT is stronger than the decidable fan theorem (DFT);
however, Loeb [Ann. Pure Appl. Logic, 132(1):51-66, 2005] has shown that the
two principles become equivalent with a suitable coding of "continuous
functions" as type-one objects. The question remains whether DFT can be
characterised by a weaker version of UCT using a natural subclass of pointwise
continuous functions without such a coding. We show that when "pointwise
continuous" is replaced with "having a continuous modulus", UCT becomes
equivalent to DFT. We also show that this weakening of UCT is equivalent to a
similar principle for real-valued functions on the Cantor space
$\{0,1\}^{\mathbb{N}}$. These results extend Berger's characterisation of DFT
by the similar principle for functions from $\{0,1\}^{\mathbb{N}}$ to
$\mathbb{N}$, and unifies these characterisations of DFT in terms of functions
having continuous moduli. Furthermore, we directly show that the continuous
real-valued functions on the unit interval having continuous moduli are exactly
those 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.
