  Let $X\subset \mathbb R^n$ be a connected locally closed definable set in an
o-minimal structure. We prove that the following three statements are
equivalent: (i) $X$ is a $C^1$ manifold, (ii) the tangent cone and the
paratangent cone of $X$ coincide at every point in $X$, (iii) for every $x \in
X$, the tangent cone of $X$ at the point $x$ is a $k$-dimensional linear
subspace of $\mathbb R^n$ ($k$ does not depend on $x$) varies continuously in
$x$, and the density $\theta(X, x) < 3/2$.
