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We give examples of $L^{1}$-functions that are essentially unbounded on everynonempty open subset of their domains of definition. We obtain such functionsas limits of weighted sums of functions with the unboundedly increasing numberof singular points lying at the nodes of standard compressible periodic gridsin $\Bbb R^n$. Moreover, we prove that the latter (basic) functions possessproperties of uniform integral boundedness but do not have a pointwisemajorant. Some applications of the main results are given.

On $L^1$-Functions with a very Singular Behavior