In a recent paper by the author, joint with O.\,G.~Fotiy and M.\,M.~Popov, we used the orthogonally additive projection of $L_0$ onto the lateral band generated by the unit function, to construct an example of a nonzero orthogonally additive functional on $L_0$, which is continuous at zero. However, all examples of the kind are discontinuous at some other points. So the question on the existence of a nonzero orthogonally additive functional on $L_0$, which is continuous at every point, naturally arises. We give an affirmative answer to this question by constructing of two different types of such examples. Next we apply the obtained results to the Arrow-Debreu model of economy.

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