  In this paper we introduce and study the so-called continuous $K$-theory for
a certain class of "large" stable $\infty$-categories, more precisely, for
dualizable presentable categories. For compactly generated categories, the
continuous $K$-theory is simply the usual (non-connective) $K$-theory of the
full subcategory of compact objects. More generally, we show that any
localizing invariant of small stable $\infty$-categories can be uniquely
extended to a localizing invariant of dualizable categories.
  We compute the continuous $K$-theory for categories of sheaves on locally
compact Hausdorff spaces. Using the special case for sheaves on the real line,
we give an alternative proof of the theorem of Kasprowski and Winges
\cite{KW19} on the commutation of $K$-theory with infinite products for small
stable $\infty$-categories.
  We also study the general theory of dualizable categories. In particular, we
give an "explicit" proof of Ramzi's theorem \cite{Ram} on the
$\omega_1$-presentability of the category of dualizable categories. Among other
things, we prove that dualizability is equivalent to "flatness" in the category
of presentable stable categories.
