For \(I=\mathbb{R}\) or \(\mathbb{R}_ +\) and \(\beta\in\mathbb{R}\), consider the function space \[ W_ \beta(I)= \Bigl\{\varphi\in C^ \infty (I\setminus\{0\}):\;p_{n,\beta} (\varphi)= \int_ I | x^{\beta+n} \varphi^{(n)}(x)| dx<\infty,\;n\in N_ 0 \Bigr\} \] endowed with the seminorms \(\{p_{n,\beta}\): \(n\in N_ 0\}\), where \(N_ 0= N\cup\{0\}\). Consider a subspace \(V\) of \(W_ \beta(I)\) such that the inclusion map is continuous as well as the map \[ V\ni \varphi(\cdot)\to \varphi(\cdot/y)\in V, \qquad y\in\mathbb{R}_ +. \] It is proved that \(f\in V'\) is homogeneous of order \(\alpha\) if and only if \(xf'= \alpha f\) (in \(V'\)). Moreover a sufficient condition for the space \(V'\) is given such that all generalized functions from \(V'\) which are homogeneous of order \(\alpha\) are of the form \(ax_ +^ \alpha+ bx_ -^ \alpha\) on \(\mathbb{R}\) or \(ax_ +^ \alpha\) on \(\mathbb{R}_ +\), where \(a,b\in\mathbb{R}\) and \(x_ \pm^ \alpha= x^ \alpha H(\pm x)\), \(H\) being the characteristic function of \(\mathbb{R}_ +\).