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

The continuity of $\chi$-volume functions over adelic curves