  In the setting of Arakelov geometry over adelic curves, we introduce the
$\chi$-volume function and show some general properties. This article is
dedicated to talk about the continuity of $\chi$-volume function. By discussing
its relationship with volume function, we prove its continuity around adelic
$\mathbb{Q}$-ample $\mathbb{Q}$-Cartier divisors and its continuity in the
trivially valued case. The study of the variation of arithmetic Okounkov bodies
leads us to its continuous extension on arithmetic surfaces.
