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We prove a version of the implicit function theorem for Lipschitz mappings$f:\mathbb{R}^{n+m}\supset A \to X$ into arbitrary metric spaces. As long asthe pull-back of the Hausdorff content $\mathcal{H}_{\infty}^n$ by $f$ haspositive upper $n$-density on a set of positive Lebesgue measure, then, thereis a local diffeomorphism $G$ in $\mathbb{R}^{n+m}$ and a Lipschitz map$\pi:X\to \mathbb{R}^n$ such that $\pi\circ f\circ G^{-1}$, when restricted toa certain subset of $A$ of positive measure, is a the orthogonal projection of$\mathbb{R}^{n+m}$ onto the first $n$-coordinates. This may be seen as aqualitative version of a similar result of Azzam and Schul. The main tool inour proof is the metric change of variables introduced in a paper of Hajlaszand Malekzadeh.

An implicit function theorem for Lipschitz mappings into metric spaces