In this note, we study the regularity of $\mathbb{R^N}$-valued random functions $X - (X_n, n \in \mathbb{N})$ on $\mathbb{R}$ such that $X_n(t) = a_nx_n(b_nt),\quad t \in \mathbb{R}, n \in \mathbb{N},$ where $\mathbf{a} = (a_n) \subset \mathbb{R}^+, \mathbf{b} = (b_n) \subset \mathbb{R}^+$ and $(x_n, n \in \mathbb{N})$ is an i.i.d. sequence of gaussian centered stationary real random functions on $\mathbb{R}$. If the common covariance of the $x_n$'s verifies some very weak regularity assumptions, then their paths are continuous in $l_p, p \in \lbrack 1,\infty\lbrack$ if and only if they are in this space and some integral depending uniquely on $p$ and on $\mathbf{a}$ and $\mathbf{b}$ is convergent. These results extend and refine some previous results concerning only the case $p \in \lbrack 2, \infty\lbrack$.