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As our main theorem, we prove that a Lipschitz map from a compact Riemannianmanifold $M$ into a Riemannian manifold $N$ admits a smooth approximation viaimmersions 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 abi-Lipschitz homeomorphism between compact manifolds and its inverse map haveno singular points in the same sense, then they are diffeomorphic. We havethree applications of the main theorem: The first two of them are twodifferentiable sphere theorems for a pair of topological spheres including thatof exotic ones. The third one is that a compact $n$-manifold $M$ is a twistedsphere 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 itsdistance function in the Clarke sense. Moreover, we have three corollaries fromthe third theorem; the first one is that for any twisted sphere $\Sigma^n$ ofgeneral dimension $n$, there exists a bi-Lipschitz homeomorphism between$\Sigma^n$ and $S^n(1)$ which is a diffeomorphism except for a single point. Inparticular, there exists such a map between an exotic $n$-sphere $\Sigma^n$ ofdimension $n>4$ 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 conditionsabove; the third one is that for any Grove-Shiohama type $n$-sphere $N$, thereexists a bi-Lipschitz homeomorphism between $N$ and $S^n(1)$ which is adiffeomorphism except for one of points that attain their diameters.

Approximations of Lipschitz maps via immersions and differentiable exotic sphere theorems