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[1] The aim of this paper is to assess the scaling properties of heavy point rainfall with respect to duration. In the region of interest, the probability distribution tails of hourly to daily rainfall display log-log linearity. The log-log linearity of tails is a feature of fat-tailed distributions. The conservation of this property throughout the scales will be investigated in the framework of scale-invariant analysis. Evidence of the scaling of heavy rainfall is shown for one particularly long rainfall series through the conservation of the survival probability shape at durations in the range 1-24 h. An objective method is implemented to estimate the hyperbolic-tail parameters of rainfall distributions. This method is automatized and detects the lower bound above which the distributions exhibit power law tails and determines the power law exponent a using a maximum likelihood estimator. The application of unbiased estimation methods and scale-invariant properties for the estimation of the power law exponent provides a significant reduction of the intergage power law variability. This achievement is essential for a correct use of geostatistical approaches to interpolate the power law parameters at ungaged sites. The method is then applied to the rain gage network in the Cévennes-Vivarais region, a Mediterranean mountainous region located in southern France. The maps show thicker rainfall distribution tails in the flat area between the seashore and the foothill. It is shown that in a flat region closer to the Mediterranean Sea the rainfall distribution tails are hyperbolic and the power law exponent is quasi-constant with duration, whereas, over the mountain, the power law behavior is less defined. The physical reasons for such results and the consequences for the statistical modeling of heavy rainfall are then discussed, providing an innovative point of view for the comprehension of the rainfall extremes behavior at different temporal scales.