  We identify a class of two-mode squeezed states which are parametrized by an
angular variable ${0\le\theta<2\pi}$ and a squeezing parameter $r$. We show
that, for a large squeezing value, these states are either (almost) maximally
entangled or product states depending on the value of $\theta$. This peculiar
behavior of entanglement is unique for infinite dimensional Hilbert space and
has consequences for the entangling power of unitary operators in such systems.
Finally, we show that, at the limit ${r\to\infty}$ these states demonstrate a
discontinuity attribute of entanglement.
