Introduction. V. V. Golubev, in his study [6] , has constructed, by using definite integrals, various examples of analytic functions having a perfect nowhere-dense set of singular points. These functions were shown to be singlevalued with a bounded imaginary part. In attempting to extend his work to the problem of constructing analytic functions having perfect, nowhere-dense singular sets under quite general conditions, he posed the following question: Given an arbitrary, perfect, nowhere-dense

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... point-set E of positive measure in the complex plane, is it possible to construct, by passing a Jordan curve through E and by using definite integrals, an example of a single-valued analytic function, which has E as its singular set, with its imaginary part bounded. In the present investigation, we shall require the set £, which is bounded and closed, to belong to the class of irregular sets of finite linear measure. 1 Hence, we wish to determine the possibility of obtaining, by using definite integrals, a function (z) having the following properties: