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Norms and derivatives. (English) Zbl 0787.26009

The paper is a continuation of J. Mařík and C. E. Weil [Proc. Am. Math. Soc. 112, No. 3, 807-817 (1991; Zbl 0746.26002)]. The aim is to study conditions under which the square root of the sum of squares of two derivatives is still a derivative. The conditions are formulated in terms of multiplicative families for bounded approximately continuous functions and stated in Theorem 6.2 (even in greater generality for \(p>0\)). The paper includes also some negative results and generalizations for \(n\)-tuples of derivatives.

MSC:

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable

Citations:

Zbl 0746.26002