  In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal
target. These graphs are $C^1_H$ regular exactly when the map is uniformly
intrinsically differentiable. Our first main result characterizes the uniformly
intrinsic differentiability by means of H\"older properties along the
projections of left-invariant vector fields on the graph. We strengthen the
result in step-2 Carnot groups for intrinsic real-valued maps by only requiring
horizontal regularity. We remark that such a refinement is not possible already
in the easiest step-3 group. As a by-product of independent interest, in every
Carnot group we prove an area-formula for uniformly intrinsically
differentiable real-valued maps. We also explicitly write the area element in
terms of the intrinsic derivatives of the map.
