Let $\Omega = [0,1]^d$ and consider $f : \Omega \to R$ Lipschitz continuous with constant 1. 

Consider the regular decomposition of $\Omega$ into $d$-dimensional cubes $\Omega_i$, $i=1 ... k^d$ with $k$ subdivision intervals along each dimension.

Consider the best approximation (in the sense of $ \| \cdot \|_\infty$) of $f$ by a function $\widehat f : \Omega \to R$, which is affine on each $\Omega_i$ and also Lipschitz continuous with constant 1 (and hence continuous). 

I'm looking for an error/approximation bound $\| f - \widehat f \|_\infty \leq ...$ in terms of the mesh-size $k$ and dimension $d$. 

EDIT: what makes this difficult for me is the requirement that $\widehat f$ is continuous. Since this is a global property, it seems very difficult to get a meaningful bound.