Real Analysis Exchange

Separately Twice Differentiable Functions and the Equation of String Oscillation

Taras Banakh and Volodymyr Mykhaylyuk

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We prove that for every separately twice differentiable function \(f:\mathbb {R}^2\to\mathbb{R}\) with that \(f''_{xx}=f''_{yy}\) there exist twice differentiable functions \(\varphi, \psi:\mathbb R\to\mathbb{R}\) such that \(f(x,y)=\varphi(x+y) + \psi(x-y)\).

Article information

Real Anal. Exchange, Volume 38, Number 1 (2012), 133-156.

First available in Project Euclid: 29 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26B05: Continuity and differentiation questions
Secondary: 35A99: None of the above, but in this section

separately differentiable functions partial differential equations


Banakh, Taras; Mykhaylyuk, Volodymyr. Separately Twice Differentiable Functions and the Equation of String Oscillation. Real Anal. Exchange 38 (2012), no. 1, 133--156.

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  • R. Baire, Sur les fonctions de variables réelles, Ann. Mat. Pura Appl., (3) (1899), 1–123.
  • A. M. Bruckner, G. Petruska, D. Preiss, and B. S. Thomson, The equation $u_xu_y=0$ factors, Acta Math. Hungar., 57(3-4) (1991), 275–278.
  • P. R. Chernoff and H. F. Royden, The Equation $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}$, Amer. Math. Monthly, 82(5) (1975), 530–531.
  • R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
  • A. K. Kalancha and V. K. Maslyuchenko, A generalization of Bruckner-Petruska-Preiss-Thomson theorem, Matematychni Studii, 11(1) (1999), 48–52.
  • V. K. Maslyuchenko, One property of partial derivatives, Ukraïn. Mat. Zh. 39(4) (1987), 529–531.
  • V. K. Maslyuchenko and V. V. Mykhaylyuk, Solving of partial differential equations under minimal conditions, Zh. Mat. Fiz. Anal. Geom., 4(2) (2008), 252–266.
  • R. D. Mauldin (Ed.), The Scottish Book. Mathematics from the Scottish Café, Birkhäuser, Boston, 1981.
  • G. P. Tolstov, On partial derivatives, Izv. Ross. Akad. Nauk Ser. Mat., 13 (1949), 425–446.
  • G. P. Tolstov, On the second mixed derivative, Mat. Sb., 24(66):1 (1949), 27–51.