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In this paper, we consider the Cauchy problem for a three-component Novikov system on the line. We give a construction of the initial data <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo>,</mo><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><msub><mi>v</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>∈</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mo>∞</mo></mrow><mrow><mi>σ</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mo>∞</mo></mrow><mi>σ</mi></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mo>∞</mo></mrow><mi>σ</mi></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>></mo><mo movablelimits="true" form="prefix">max</mo><mfenced separators="" open="{" close="}"><mrow><mn>3</mn><mo>+</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>,</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></mfenced><mo>,</mo><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, such that the corresponding solution to the three-component Novikov system starting from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo>,</mo><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><msub><mi>v</mi><mn>0</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula> is discontinuous at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> in the metric of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mo>∞</mo></mrow><mrow><mi>σ</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mo>∞</mo></mrow><mi>σ</mi></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mo>∞</mo></mrow><mi>σ</mi></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, which implies the ill-posedness for this system in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mo>∞</mo></mrow><mrow><mi>σ</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mo>∞</mo></mrow><mi>σ</mi></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mo>∞</mo></mrow><mi>σ</mi></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>.