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.