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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
Keywords:
uniform limit point