  Let $G$ be a connected general graph of even order, with a function $f\colon
V(G)\to\Z^+$. We obtain that $G$ satisfies the Tutte's condition \[ o(G-S)\le
\sum_{v\in S}f(v)\qquad\text{for any nonempty set $S\subset V(G)$}, \] with
respect to $f$ if and only if $G$ contains an $H$-factor for any function
$H\colon V(G)\to 2^\N$ such that $H(v)\in \{J_f(v),\,J_f^+(v)\}$ for each $v\in
V(G)$, where the set $J_f(v)$ consists of the integer $f(v)$ and all positive
odd integers less than $f(v)$, and the set $J^+_f(v)$ consists of positive odd
integers less than or equal to $f(v)+1$. We also obtain a characterization for
graphs of odd order satisfying the Tutte's condition with respect to a
function.
