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For continuous iteration index {{mvar|t}}, then, now written as a subscript, this amounts to Lie's celebrated exponential realization of a continuous group,
:<math>e^{t~\frac{\partial ~~}{\partial h(x)}} g(x)= g(h^{-1}(h(x )+t))= g(f_t(x)).</math>
The initial flow velocity {{mvar|v}} suffices to determine the entire flow, given this exponential realization which automatically provides the general solution to the ''translation functional equation'',<ref name="acz">Aczel, J. (2006), ''Lectures on Functional Equations and Their Applications'' (Dover Books on Mathematics, 2006), Ch. 6, {{ISBN|978-0486445236}}.</ref>
:<math>f_t(f_\tau (x))=f_{t+\tau} (x) ~.</math>
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