On the topology of the Reeb graph

The Reeb quotient space ${R_f}$ of a function $f: X\to\mathbb{R}$, known as the Reeb graph, can have various properties depending on $X$ and $f$. In the classical case of a smooth function on a closed manifold with a finite number of critical points, ${R_f}$ has the structure of a finite graph. Recently, Saeki showed that the same is true if $f$ is a smooth function with a finite number of critical values. Expanding his result, we prove that for an arbitrary smooth function on a closed connected manifold, the Reeb space ${R_f}$ still has a "good'' structure; namely, ${R_f}$ is a 1-dimensional Peano continuum homotopy equivalent to a finite graph.