Symmetric differentiability of functions of two variables
A function \(f: \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) is said to be symmetrically differentiable at the point \((x_0, y_0)\) if there exist finite numbers \(A= A(x_0, y_0)\) and \(B= (x_0, y_0)\) such that the equality NEWLINE\[NEWLINE\lim_{(h,k)\to (0,0)} (f(x_0+ h,y_0+ k)- f(x_0- h,y_0- k)- 2Ah- 2Bk)/(|h|+|k|)= 0NEWLINE\]NEWLINE is fulfilled. The author investigates this notion in connection with notions of symmetric partial derivatives in usual, in strong and in the angular sense. E.g., it is proved: If there exist symmetric strong partial derivatives \(f^{(')}_{(1)}(x_0, y_0)\) and \(f^{(')}_{(2)}(x_0, y_0)\), then \(f(x,y)\) is symmetrically differentiable at \((x_0, y_0)\) and not vice versa (a function \(f(x,y)\) is said to possess at the point \((x_0, y_0)\) a symmetric strong partial derivative with respect to \(x\) if there exists the limit NEWLINE\[NEWLINE\lim_{(h,k)\to (0,0)} (f(x_0+ h,y_0+ k)- f(x_0- h,y_0+ k))/2h= f^{(')}_{(1)}(x_0, y_0);NEWLINE\]NEWLINE the symmetric strong partial derivative with respect to \(y\), \(f^{(')}_{(2)}(x_0, y_0)\), is defined analogously).
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