Changing the order of partial differentiation
Let f(x,y) be a real function defined on an open subset U of \(R^ 2\). Let \(D_ 1f\) \((D_ 2f)\) denote the partial derivative of f with respect to the 1-st place (2-nd place) variable. Also \(D_ 1D_ 2f\) denotes the derivative \(D_ 1(D_ 2f)\) of the function \(D_ 2f\), etc. Let \(D^+_ 1f\) \((D^+_ 2f)\) denote de upper right Dini derivate of f with respect to the 1-st place (2-nd place) variable. Theorem 1. Let \(D_ 1f\), \(D_ 2f\), \(D_ 1D_ 1f\), \(D_ 2D_ 2f\) exist and be finite on U. If \(D^+_ 1D_ 2D_ 2f\) is locally bounded at some point (x,y)\(\in U\) (i.e., it is bounded in some neighborhood of (x,y)), then \(D_ 1D_ 2f\) and \(D_ 2D_ 1f\) exist and are finite at (x,y), and \(D_ 1D_ 2f(x,y)=D_ 2D_ 1f(x,y).\) Theorem 2. Let \(D_ 1f\), \(D_ 2f\), \(D_ 1D_ 1f\), \(D_ 2D_ 2f\) exist and be finite on U and let \(D_ 1D_ 1f\) and \(D_ 2D_ 2f\) be continuous at some point (x,y)\(\in U\). If either \(D_ 1D_ 2f\) or \(D_ 2D_ 1f\) exists at (x,y) (finite or not), then so does the other and \(D_ 1D_ 2f(x,y)=D_ 2D_ 1f(x,y).\) Theorem 3. Let f be continuous on U, and let \(D_ 1f\), \(D_ 2f\), \(D_ 1D_ 1f\) and \(D_ 2D_ 2f\) exist and be finite on U. Then the set \(\{(x,y)\in U:\) one of the derivatives \(D_ 1D_ 2f(x,y),\) \(D_ 2D_ 1f(x,y)\) exists but the other does not, or they both exist and \(D_ 1D_ 2f(x,y)\neq D_ 2D_ 1f(x,y)\}\) is a first category subset of \(R^ 2.\) Remark of the reviewer: For alternative results see \textit{A. E. Currier} [Trans. Am. Math. Soc. 35, 245-253 (1933; Zbl 0006.19403)], \textit{J. Geffroy} [Acad. Roy. Belg. Bull. Cl. Sci., V. Sér. 42, 533-542 (1956; Zbl 0074.287)], \textit{G. P. Tolstov} [Mat. Sbornik, II. Ser. 24(66), 27-51 (1949; Zbl 0045.333)] and the reviewer [C. R. Acad. Sci. Paris, Ser. A 246, 522- 524 (1958; Zbl 0078.046)].
change of order of partial differentiation
scientific article; zbMATH DE number 4142360