On plane ring $$\omega $$-homeomorphisms

Abstract We show that under some additional conditions, a plane ring $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -homeomorphism satisfies a generalized Poletsky modular inequality. In this way we can prove some results concerning the extension of continuity and the modulus of continuity around some “thin” exceptional sets.