Summary.
We prove that - under certain conditions - measurable solutions $f$ of the functional equation $f(x)=h(x,y,f(g_{1}(x,y)),\ldots,f(g_{n}(x,y))),\quad(x,y)\in D \subset \mathbb{R}^{s} \times \mathbb{R}^{l}$ are continuous, even if $1\le l\le s$. As a tool we introduce new classes of functions which - roughly speaking - interpolate between continuous and Lebesgue measurable functions. Connection between these classes are also investigated.
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Járai, A. Measurability implies continuity for solutions of functional equations - even with few variables . Aequ. math. 65, 236–266 (2003). https://doi.org/10.1007/s00010-003-2666-x
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DOI: https://doi.org/10.1007/s00010-003-2666-x