Let $P,Q$ be posets and endow them with the [interval topology][1] $\tau_i(P)$ and $\tau_i(Q)$ respectively. Is it true that if $f: P\to Q$ is order-preserving, then it is continuous, and vice versa?

  [1]: https://mathoverflow.net/questions/215673/properties-of-the-interval-topology-of-the-lattice-of-functions