Bi-Lipschitz decomposition of Lipschitz functions into a metric space

We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can be decomposed f into a finite number of BiLipschitz functions f vertical bar(Fi) so that the k-Hausdorff content of f([0, 1](k)\boolean OR F(i)) is small. We thus generalize a theorem of P. Jones [7] from the setting of R(d) to the setting of a general metric space. This positively answers problem 11.13 in "Fractured Fractals and Broken Dreams" by G. David and S. Semmes, or equivalently, question 9 from "Thirty-three yes or no questions about mappings, measures, and metrics" by J. Heinonen and S. Semmes. Our statements extend to the case of coarse Lipschitz functions.