How can a particle be unbounded by its energy?

Considering a 1D potential energy function $U(x)$, if the total energy $E$ of a particle is greater than $U(x)$ for values of $x$ going to infinity, then if left alone at some $x$, it will just move forever. It might slow down or speed up according to its kinetic energy $T=E-U(x)$ changing with $x$, but it will keep going forever either left or right along the $x$ axis according to its initial velocity direction.

In contrast, if in some region of $x$, we have $E>U(x)$ between points such that $E=U(x_a)$ and $E=U(x_b)$, then the particle will oscillate between the ``turning points'' $x_a$ and $x_b$. The particle won't be able to escape the region; it is said to be ``bound''. A particle can't exist with $E<U(x)$ since $T$ must be positive.