In this paper, we study a continuity of the “values” of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer structure of the continuity of these values with respect to continued fraction expansions and it turns out that it is different from the continuity with respect to Euclidean topology.

中文翻译：

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模块化函数的循环积分的连续性

在本文中，我们研究了实函数二次函数中模块函数“值”的连续性，这些实函数是根据沿相关封闭测地线的循环积分定义的。我们的主定理揭示了相对于连续分数展开的这些值的连续性的更精细结构，结果证明，它与关于欧几里得拓扑的连续性不同。