We know that the necessary and sufficient conditions for a multivariate function to be differentiable at a certain point are complicated:
Suppose the function $z = f (x_ 1, x_ 2, ..., x_n) $ is defined in the neighborhood $U$ of the point $P_ 0 (x_ {10}, x_ {20}, ..., x_{n0}) $. Then the sufficient and necessary conditions for the function $z = f (x_ 1, x_ 2, ..., x_n) $ to be differentiable at the point $P_ 0 (x_ {10}, x_ {20}, ..., x_{n0}) $ are:
The n first-order partial derivatives of the function $z = f (x_ 1, x_ 2, ..., x_n) $ at the point $P_ 0 (x_ {10}, x_ {20}, ..., x_{n0}) $ all exist, and $$f (x_ 1, x_ 2, ..., x_n) - f (x_ {10}, x_ 2, ..., x_n) - f (x_ 1, x_ {20}, ..., x_n) - ... -f (x_ 1, x_ 2, ..., x_{n0}) + f (x_ {10}, x_ {20}, ..., x_{n0}) = o (\rho) $$
where $(x_ 1, x_ 2, ..., x_n) \in U$, $\rho = \sqrt{(x_ 1 - x_ {10})^2 + (x_ 2 - x_ {20})^2 + ... + (x_n - x_ {n0})^2}$.
I already know that the following bivariate function $f(x,y)$ is differentiable at point $(0,0)$, but its two first-order partial derivatives are not continuous at $(0,0)$:
$$f(x, y)=\begin{cases}(x^2 + y^2) \sin(\frac{1}{(x^2 + y^2)}), &(x, y) \neq (0, 0) \cr 0 , &(x, y)=(0, 0)\end{cases} $$
f[x_, y_] :=
Piecewise[{{(x^2 + y^2) Sin[1/(x^2 + y^2)], x^2 + y^2 != 0}}, 0]
D[f[x, y], x] /. {x -> 0, y -> 0}
D[f[x, y], y] /. {x -> 0, y -> 0}
Limit[(f[x, y] - f[x, 0] - f[0, y] + f[0, 0])/Sqrt[
x^2 + y^2], {x, y} -> {0, 0}]
I want to write a custom function to judge whether a bivariate function is differentiable at a certain point. How should I write this function?
For example, through this custom function, we'll be able to judge that the following bivariate function is NOT differentiable at $(0,0)$:
$$f(x, y)=\begin{cases}\frac{x^2y}{x^4 + y^2}, &(x, y) \neq (0, 0) \cr 0, &(x, y)=(0, 0)\end{cases} $$
The following bivariate function should be differentiable at point $(0,0)$:
$$f(x, y)=\begin{cases}(x^2 + y^3) \sin(\frac{1}{(x^2 + y^2)}), &(x, y) \neq (0, 0) \cr 0, &(x, y)=(0, 0)\end{cases} $$
f[x_, y_] :=
Piecewise[{{(x^2 + y^3) Sin[1/Sqrt[x^2 + y^2]], x^2 + y^2 != 0}}, 0]
Limit[(f[0 + Δx,
0 + Δy] - (D[f[x, y], x] /. {x -> 0,
y -> 0}) Δx - (D[f[x, y], y] /. {x -> 0,
y -> 0}) Δy)/Sqrt[Δx^2 + Δy^2], {Δx, Δy} -> {0, 0}]
Correction information:
After careful examination, I found that the theorem in the paper was wrongly written due to the author's negligence. The correct form is as follows:
$$f (x_ 1, x_ 2, ..., x_n) - f (x_ 1, x_ {20}, ..., x_ {n0}) - f (x_ {10}, x_ 2, ..., x_ {n0}) - ... - f (x_ {10}, x_ {20}, ..., x_n) + (n - 1) f (x_ {10}, x_ {20}, ..., x_ {n0}) = o (\rho)$$
where n is the number of variables of this multivariate function.
