When does restricted continuity on continuous function graphs imply joint continuity?
When does restricted continuity on continuous function graphs imply joint continuity?
We generalize a result of Luzin about real functions of real variables —that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x comma y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">F(x,y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is jointly continuous if and only if <inline-formula content-type="math/mathml"> <mml:math …