Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class Γ , then its Γ -code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau's theorem to Borel functions: If a Borel function on a Polish space happens to be a Σ ~ t -function, then one can find its Σ ~ t -code hyperarithmetically relative to its Borel code. More generally, we prove extension-type, domination-type, and decomposition-type variants of Louveau's theorem for Borel functions. [ABSTRACT FROM AUTHOR]