Closure of Deterministic context-free languages under prefix For a formal language $L \subseteq \Sigma^{*}$ I define the set Pref(L) to be:

$\text{pref}(L) = \{\alpha \in \Sigma^{*} : \exists \beta \in \Sigma^{*} \text{ such that } \alpha \beta \in L\}$

ie. the set of all (not necessarily proper) prefixes of words in $L$. I know that if $L$ is context-free then pref(L) is context-free but if $L$ is deterministic context-free then is pref(L) deterministic context-free?

I am sure this is known but I cannot find the answer anywhere and it's not in Hopcroft and Ullman.
DCFL are known to be closed under quotient with regular languages, but the quotient of $L$ with $\Sigma^{*}$ is precisely $\text{pref}(L)$ so yes, if $L$ is a DCFL then $\text{pref}(L)$ is a DCFL.