  As our main theorem, we prove that a Lipschitz map from a compact Riemannian
manifold $M$ into a Riemannian manifold $N$ admits a smooth approximation via
immersions if the map has no singular points on $M$ in the sense of F.H.
Clarke, where $\dim M \leq \dim N$. As its corollary, we have that if a
bi-Lipschitz homeomorphism between compact manifolds and its inverse map have
no singular points in the same sense, then they are diffeomorphic. We have
three applications of the main theorem: The first two of them are two
differentiable sphere theorems for a pair of topological spheres including that
of exotic ones. The third one is that a compact $n$-manifold $M$ is a twisted
sphere and there exists a bi-Lipschitz homeomorphism between $M$ and the unit
$n$-sphere $S^n(1)$ which is a diffeomorphism except for a single point, if $M$
satisfies certain two conditions with respect to critical points of its
distance function in the Clarke sense. Moreover, we have three corollaries from
the third theorem; the first one is that for any twisted sphere $\Sigma^n$ of
general dimension $n$, there exists a bi-Lipschitz homeomorphism between
$\Sigma^n$ and $S^n(1)$ which is a diffeomorphism except for a single point. In
particular, there exists such a map between an exotic $n$-sphere $\Sigma^n$ of
dimension $n>4$ and $S^n(1)$; the second one is that if an exotic $4$-sphere
$\Sigma^4$ exists, then $\Sigma^4$ does not satisfy one of the two conditions
above; the third one is that for any Grove-Shiohama type $n$-sphere $N$, there
exists a bi-Lipschitz homeomorphism between $N$ and $S^n(1)$ which is a
diffeomorphism except for one of points that attain their diameters.
