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At $z=x+iy$, compute the complex derivative first along real increments and then along imaginary increments. Holomorphy givesEquating real and imaginary parts proves the Cauchy-Riemann equations:The partial derivatives exist because these directional difference quotients are limits of the holomorphic derivative. This argument proves the necessary equations; it does not claim that their pointwise existence alone is sufficient for holomorphy.

Take $M=\mathbb{R}$ and the constant contraction mapping $f(x)=0$. For $p=(0,0)$, $q=(0,1)$,No Lipschitz constant less than one can work, so $F$ need not be a contraction. More generally, differing only in the second input coordinate already prevents a strict contraction whenever $M$ has two points.

If $f$ has contraction constant $0\le c<1$, two iterations giveand consequentlyThus $F^2$ is a contraction mapping. The swap lift of a contraction shows why an iterate can contract even when one step does not.

For coordinate differences $a,b$, the elementary inequalitiesgiveThe identity map between the two versions of the square is bijective, and both it and its inverse are Lipschitz continuous maps by these inequalities. It is therefore a homeomorphism. In fact the same comparison works on all of ${\mathbb{R}}^2$; restriction to the square causes no difficulty.