| Summary: | Symmetry is repetitive self-similarity. We proved the stability problem by replicating the well-known Cauchy equation and the well-known Jensen equation into two variables. In this paper, we proved the Hyers-Ulam stability of the bi-additive functional equation <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><mi>y</mi><mo>,</mo><mi>z</mi><mo>+</mo><mi>w</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the bi-Jensen functional equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>4</mn><mi>f</mi><mfenced separators="" open="(" close=")"><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mspace width="0.166667em"></mspace><mfrac><mrow><mi>z</mi><mo>+</mo><mi>w</mi></mrow><mn>2</mn></mfrac></mfenced><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>z</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>w</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>z</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>w</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula>
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