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Let <i>f</i> and <i>g</i> be two continuous functions. In the present paper, we put forward a method to calculate the lower and upper Box dimensions of the graph of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow></semantics></math></inline-formula> by classifying all the subsequences tending to zero into different sets. Using this method, we explore the lower and upper Box dimensions of the graph of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow></semantics></math></inline-formula> when the Box dimension of the graph of <i>g</i> is between the lower and upper Box dimensions of the graph of <i>f</i>. In this case, we prove that the upper Box dimension of the graph of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow></semantics></math></inline-formula> is just equal to the upper Box dimension of the graph of <i>f</i>. We also prove that the lower Box dimension of the graph of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow></semantics></math></inline-formula> could be an arbitrary number belonging to a certain interval. In addition, some other cases when the Box dimension of the graph of <i>g</i> is equal to the lower or upper Box dimensions of the graph of <i>f</i> have also been studied.