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Let $\left\{X\right(t),t\in [0,1{]}^n\}$ be a stochastically continuous, separable, Gaussian process with $E\left[X\right(t+h)-X(t){]}^2={\sigma }^2(\left|h\right|)$. A sufficient condition, in terms of the monotone rearrangement of $\sigma$, is obtained for $X\left(t\right)$ to have continuous sample paths almost surely. This result is applied to a wide class of random series of functions, in particular, to random Fourier series.