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This article is a study on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula>-Riemann–Liouville fractional integral, a generalization of the Riemann–Liouville fractional integral. Firstly, we introduce several properties of the extended integral of continuous functions. Furthermore, we make the estimation of the Box dimension of the graph of continuous functions after the extended integral. It is shown that the upper Box dimension of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula>-Riemann–Liouville fractional integral for any continuous functions is no more than the upper Box dimension of the functions on the unit interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>=</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>, which indicates that the upper Box dimension of the integrand <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> will not be increased by the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula>-Riemann–Liouville fractional integral <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow></mrow><mi>k</mi><mi>s</mi></msubsup><msup><mi>D</mi><mrow><mo>−</mo><mi>σ</mi></mrow></msup><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> on <i>I</i>. Additionally, we prove that the fractal dimension of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow></mrow><mi>k</mi><mi>s</mi></msubsup><msup><mi>D</mi><mrow><mo>−</mo><mi>σ</mi></mrow></msup><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of one-dimensional continuous functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> is still one.