We identify bar recursion on moduli of continuity as a fundamental notion of constructive mathematics. We show that continuous functions from the Baire space \({{\mathbb {N}}}^{{\mathbb {N}}}\) to the natural numbers \({\mathbb {N}}\) which have moduli of continuity with bar recursors are exactly those functions induced by Brouwer operations. The connection between Brouwer operations and bar induction allows us to formulate several continuity principles on the Baire space stated in terms of bar recursion on continuous moduli which naturally characterise some variants of bar induction. These principles state that a certain kind of continuous function from \({{\mathbb {N}}}^{{\mathbb {N}}}\) to \({\mathbb {N}}\) admits a modulus of continuity with a bar recursor. The results for the Baire space are recast in the setting of the Cantor space \({{\{ 0,1 \}}}^{{\mathbb {N}}}\) using the notion of fan recursor, a bar recursor for binary trees. This yields characterisations of uniformly continuous functions on the Cantor space and fan theorem in terms of fan recursors. The results for the Cantor space hold over the extensional version of intuitionistic arithmetic in all finite types (\({\mathsf {E}}\text {-}\mathsf {HA}^\omega \)), and those for the Baire space hold over \({\mathsf {E}}\text {-}\mathsf {HA}^\omega \) extended with the type for Brouwer operations. Our work places Spector’s bar recursion in a proper context of Brouwer’s intuitionistic mathematics and clarifies the connection between bar recursion and bar induction.

我们将连续模的条递归确定为构造数学的基本概念。我们证明了从Baire空间\（{{\ mathbb {N}}} \}到具有模数的自然数\（{\ mathbb {N}} \\）的连续函数条形递归的连续性恰好是由Brouwer操作引起的那些功能。Brouwer操作与条形感应之间的联系使我们能够在Baire空间上制定一些连续性原理，这些连续性原则是对连续模量上的条形递归进行自然描述的条形感应的一些变体。这些原则指出从\（{{\ mathbb {N}}} \）到\（{\ mathbb {N}} \\）允许使用条形递归的连续模数。使用扇形递归（条形递归）概念在Cantor空间\（{{\ {0,1 \}}} ^ {{\\ mathbb {N}}} \）的设置中重铸Baire空间的结果用于二叉树。这样就得出了在Cantor空间上的一致连续函数的特征，并根据扇形递归确定了扇形定理。Cantor空间的结果保留了所有有限类型（\（{\ mathsf {E}} \ text {-} \ mathsf {HA} ^ \ omega \）的直觉算术的扩展版本，以及Baire的结果空间缓缴\（{\ mathsf {E}} \ {文本- } \ {mathsf HA} ^ \欧米茄\）扩展了用于Brouwer操作的类型。我们的工作将Spector的条形递归置于Brouwer直觉数学的适当上下文中，并阐明了条形递归与条形归纳之间的联系。