Tangent cones and C1 regularity of definable sets

Let X⊂Rn be a connected locally closed set which is definable in an o-minimal structure. We prove that the following three statements are equivalent: (i) X is a C1 manifold, (ii) the tangent cone and the paratangent cone of X coincide at every point in X, (iii) for every x∈X, the tangent cone of X at the point x is a k-dimensional linear subspace of Rn (k does not depend on x) varies continuously in x, and the density θ(X,x)<3/2.