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On points of uniform and non-uniform limits. (English) Zbl 0639.26010
Let (X,d) and (H,\(\rho)\) be metric spaces, (X,d) complete, Y a Banach space, \(\vartheta\in H\), \(J\subset H-\{\vartheta \}\) have \(\vartheta\) for limit point, \(\phi: X\times H\to Y,\phi\) (x,h) continuous on X for every fixed \(h\in J\), \(f: X\to Y\), and suppose that, for \(x\in X\) and \(\epsilon >0\), there is \(\delta >0\) such that \((1)\quad \| \phi (x,h)-f(x)\| <\epsilon\) for \(h\in J\), \(\rho (\vartheta,h)<\delta\). \(\alpha\in X\) is said to be a uniform limit point if, for \(\epsilon >0\), there are \(\delta >0\) and \(\eta >0\) such that (1) holds whenever \(d(\alpha,x)<\eta,\) \(h\in J\), \(\rho (\vartheta,h)<\delta.\)
The main result of the paper claims that the points of non-uniform limit constitute a set of first category in X. Results of B. K. Lahiri [Am. Math. Mon. 67, 649-652 (1960; Zbl 0100.055)], S. N. Mukhopadhyay [Rev. Roum. Math. Pures Appl. 9, 859-862 (1964; Zbl 0143.074)] and E. W. Hobson [Theory of functions of a real variable and of Fourier series, Vol. I (1958; Zbl 0081.277)] are deduced.
Reviewer: Á.Császár
MSC:
26B05 Continuity and differentiation questions
54C30 Real-valued functions in general topology
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