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The present paper extends the earlier results by [Abramov, V. M. Conditionsfor recurrence and transience for time-inhomogeneous birth-and-death processes,\emph{Bull. Aust. Math. Soc.} (to appear)] for the case of time-inhomogeneousrandom walks, the increments of which take values in $\mathbb{R}$. By this, wegive a full solution of the open problem formulated by [Menshikov, M. andVolkov, S. Urn-related random walk with drift $\rho x^\alpha/t^\beta$,\emph{Electron. J. Probab.}, \textbf{13} (2008), paper No. 31, 944--960] thatwas partially solved in the aforementioned paper by Abramov.

Conditions for recurrence and transience for time-inhomogeneous random walks