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This paper considers the relationship of box dimension between a continuous fractal function and its Riemann–Liouville fractional integral. For an arbitrary fractal function [Formula: see text] it is proved that the upper box dimension of the graph of Riemann–Liouville fractional integral [Formula: see text] does not exceed the upper box dimension of [Formula: see text], i.e. [Formula: see text]. This estimate shows that [Formula: see text]order Riemann–Liouville fractional integral [Formula: see text] does not increase the fractal dimension of the integrand [Formula: see text], which means that Riemann–Liouville fractional integration does not decrease the smoothness at least that is obvious known result for classic integration. Our result partly answers fractal calculus conjecture in [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 2 (1995) 217–229] and [Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of one-dimensional continuous functions, Sci. Sin. Math. 4 (2016) 423–438].

In the present paper, a one-dimensional continuous function of unbounded variation on the interval [0, 1] has been constructed. Box dimension of this function has been proved to be 1. Furthermore, Box dimension of its Riemann-Liouville fractional integral of any order has also been proved to be 1.

Let [Formula: see text] be [Formula: see text]-Hölder continuous on [Formula: see text] and well-defined about the Weyl fractional integral. Then, [Formula: see text] where [Formula: see text] and [Formula: see text]. This estimation shows that the Box dimension of [Formula: see text] leads to some similar linear dimension decrease like the Riemann–Liouville fractional integral [Y. S. Liang and W. Y. Su, Fractal dimensions of fractional integral of continuous functions, Acta Math. Sin. 32 (2016) 1494–1508].

A set of organic–rich shales of the Upper Permian Longtan Formation, which is widely developed in the northeastern part of the Sichuan Basin, is a key formation for the next step of exploration and development. At present, most studies on this set of formations have focused on the reservoir characteristics and reservoir formation mechanism of the shales, and basic studies on the palaeoenvironment and organic matter (OM) enrichment mechanism have not been fully carried out. In this paper, we recovered the sedimentary palaeoenvironment by mineralogical, elemental geochemical and organic geochemical analyses, and explored the enrichment mechanism of OM under the constraints of palaeoenvironmental evolution. The shales can be divided into two stages of sedimentary evolution: compared with the shales of the Lower Longtan Formation, the shales of the Upper Longtan Formation are relatively rich in quartz, poor in clay and carbonate minerals, and the OM type changes from type III to type II2. The depositional environment has undergone a change from sea level rise, from warm and wet climate to dry and cold climate, and from oxygen–poor condition restricted to open reduction environment; the land source input has decreased, the siliceous mineral content has increased, the biological productivity has improved, and the deposition rate has changed from high to low. A depositional model was established for the shales of the Longtan Formation, reflecting the differential reservoir formation pattern of organic matter. For the Lower Longtan Formation shales, the most important factors controlling OM content are terrestrial source input and deposition rate, followed by paleoclimate and paleo‐oxygen conditions. For the Upper Longtan Formation shales, the most important controlling factor is paleoproductivity, followed by sedimentation rate. The depositional model constructed for the Upper and Lower Longtan Formation shales can reproduce the enrichment of organic matter and provide a basis for later exploration and development.

Since fractal functions are widely applied in dynamic systems and physics such as fractal growth and fractal antennas, this paper concerns fundamental problems of fractal continuous functions like cardinality of collection of fractal functions, box dimension of summation of fractal functions, and fractal linear space. After verifying that the cardinality of fractal continuous functions is the second category by Baire theory, we investigate the box dimension of sum of fractal continuous functions so as to discuss fractal linear space under fractal dimension. It is proved that the collection of 1-dimensional fractal continuous functions is a fractal linear space under usual addition and scale multiplication of functions. Particularly, it is revealed that the fractal function with the largest box dimension in the summation represents a fractal dimensional character whenever the other box dimension of functions exist or not. Simply speaking, the fractal function with the largest box dimension can absorb the other fractal features of functions in the summation.

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