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Mathematical analysis is incomprehensible here. Please help. Thank you.
Hello, this question involves mathematical analysis and logic.

The first part is the differential process of binary function.

Let's review the unit function differential DELTAY=`=DY=F'X *DELTA X+O(X), that is, replace the infinitesimal with the function value in the x direction.

So what? The differential process of binary function, of course, is to replace tiny quantities with function values in X and Y directions.

The second part, partial derivative continuity.

At the middle point of the graph, both partial derivatives exist at that point, but because the partial derivatives are discontinuous (that is, if X is a variable, the function values of the partial derivative function of X can be continuously connected regardless of the value), when X or Y in the graph is relatively large, the partial derivatives obviously do not exist, so there is no continuity.