The non-existence of a Hamel-basis and the general solution of Cauchy's functional equation for nonnegative numbers

with an arbitrary constant c . As DARBOUX has proved ([4]), this (with nonnegative c's) is also the most general solution of (1) which is nonnegative for positive -Ys (it is even enough to suppose the nonnegativity for small positive x's) . But without any regularity-suppositions (2) isn't anymore the most general solution of (I) . this can be shown and at the same time the most general solution of (1) can be constructed with the Hamel-basis of real numbers ([7]) . In all these results (I) was supposed valid for all real x, y and then also (2) is verified for all real x's moreover, the Hamel-basis also gives a representation of all real numbers. But for many applications (see e . g . [l]), (1) can be supposed valid only for nonnegaiire x, y . It is easy to show that (2) (with nonnegative x) remains the most general continuous solution of (I) also with this restriction and with nonnegative x, c also the most general solution nonnegative for (small) positive variables . But how to construct in this case the most general solution of (1) for nonnegatii e x, y? Are there Hamel-bases of the nonnegatit -e numbers? In this little note we answer the second question in the negative and construct nevertheless the general solution asked for in the first one .