I was trying to prove the following statement:
Let $\mathcal{F}$ be the Reeb foliation of $S^3$ and let $\phi$ : $S^3$ $\rightarrow$ $N$ be a continuous map whose restriction to each leaf of $\mathcal{F}$ is constant. Show that $\phi$ is constant.
Considering that the Reeb foliation of the 3-sphere comes from the foliation of two solid tori where the inner leaves approach the boundary (which is also a leaf) of the torus asymptotically, my idea was:
Consider a point $p$ on the boundary of the torus. Now, $\phi$($p$)=$q$. Take an open set U around $q$. $\phi^{-1}(U)$ will be an open set around $p$ and since the inner leaves tend asymptotically to the boundary of the torus, this neighborhood of $p$ will contain points from other leaves. We can make $U$ as small as we wish and the same thing will still happen. Therefore, the function will have the same value on every leaf and so it is constant on the whole manifold. I'm having some difficulty in making precise the fact that the leaves will always tend to points on the boundary, but it seems obvious from the parameterization I'm looking at: \begin{equation} (x,y) \rightarrow (x,y,log \left( \frac{c}{f(x^2+y^2)}\right)) \end{equation} with $c<0$ and $f(0)=-1$, $f(1)=0$ and $f$ is an increasing function. Note that this parameterization corresponds to leaves inside a cylinder. Then after taking the quotient it induces a foliation on the solid 2-torus.