2. An outlier is a point whose domain is completely outside the given point set, so it is called an outlier.
3. A boundary point refers to any domain that is a point, and there are both external points and internal points in the domain, so it is called a boundary point; Convergence point is a unified definition of boundary point and interior point.
4. The point set of an open set refers to all interior points.
5. A closed set refers to a point in a set that has both interior points and boundary points.
6. Connected set can be intuitively understood as an independent point set that has not been divided.
7. An independent point set that is not divided is also an open set, so it becomes a region or an open region.
8. Independent points that are inseparable and closed sets are integrated into closed regions.
9. Bounded set can be understood as a finite set of points.
Extended data:
A summary of multivariate function differential method
1, limit existence condition
The existence of limit means that when P(x, y) approaches P0(x0, y0) in any way, the function approaches A infinitely. If P(x, y) approaches P0(x0, y0) in some way, for example, even if the function approaches a certain value infinitely, we cannot draw the conclusion that the function limit exists. On the other hand, if the function tends to different values when P(x, y) tends to P0(x0, y0) in different ways, it can be concluded that the limit of this function does not exist.
For example, the function: f (x, y) = {0 (xy)/(x 2+y 2) x 2+y 2 ≠ 0}
2. Continuity
(1) definition Let the function f(x, y) be defined in the open region (or closed region) d, where P0(x0, y0) is the inner point or boundary point of d, and P0∈D, if lim (x → x0, y → y0) f(x, y) = f.
(2) A multivariate continuous function whose properties (maximum theorem and minimum theorem) are in a bounded closed region D must have a maximum value and a minimum value on D. ..
(3) A multivariate continuous function whose property (the mean value theorem) is in a bounded closed region D. If two different function values are obtained on D, it will obtain any value between these two values at least once on D..
3. Continuity and differentiability
If a univariate function has a derivative at a certain point, it must be continuous at that point, but for multivariate functions, even if all partial derivatives exist at a certain point, there is no guarantee that the function is continuous at that point. This is because the existence of partial derivatives can only ensure that the function value f(P) tends to f(P0) when the point p tends to P0 in the direction parallel to the coordinate axis, but it cannot guarantee that the function value f(P) tends to f(P0) when the point p tends to P0 in any way.
4. Necessary conditions for differentiability
The existence of derivative of univariate function at a certain point is a necessary and sufficient condition for the existence of differential, but the existence of partial derivative of multivariate function is only a necessary condition for the existence of total differential, not a sufficient condition, that is, differential = >; Differentiable
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