Fix a C^0 automorphism f of R^n, a real number r.

Then f({x_n=r}) is a topological n-1 submanifold of R^n.

I ask if there is a real number s, a continuous function F : R^{n-1} x [0,1] ---> R^n fulfilling all conditions below:

F(x,0)= f(x) for any x \in {x_n=r};

F(x, 1) \in {x_n=s};

F is an homeomorphism between R^{n-1} x [0,1] and its image in R^n.

What if f is an homeomorphism between the closure of open subsets of R^n?