The continuity of multivariate functions is neither a sufficient condition nor a necessary condition for the existence of partial derivatives.

The continuity of partial derivatives is a stronger condition, that is, the existence of partial derivatives and continuity can lead to the continuity of multivariate functions, and vice versa.

The following analysis, first of all, we need to understand that these definitions are defined by people themselves and can reflect some characteristics of multivariate functions. Therefore, as long as we grasp the meaning of these definitions, we can see the essence behind them and judge the relationship between them.

Multivariate function can be differentiated at a certain point, but it may have different limits in different directions at that point, so it is not necessarily continuous.

Whether the partial derivative of continuous function must exist is also very common in unary function. For example, when x=0, the absolute value of x has no derivative.

Partial derivative continuity (that is, partial derivative continuity! Not the existence of partial derivative+functional continuity! That is, the partial derivative exists and the partial derivative is continuous), it can be deduced that it is differentiable.

Differentiability is a strong conclusion, because if it can be approximated by a very special linear function, then many special counterexamples will disappear, and the linear function is continuous, which can be seen from the definition.

Therefore, if the partial derivative exists and is continuous, it can be deduced that the function is continuous, and vice versa.

The counterexample follows the previous counterexample, the function is continuous, but the partial derivative does not exist.

Extended data:

Partial derivative in x direction

There is a binary function z=f(x, y), and the point (x0, y0) is a point on its domain d, with y fixed at y0 and x at x0? There is an increment △x, and accordingly the function z=f(x, y) also has an increment (called partial increment to x) △z=f(x0+△x, y0)-f(x0, y0).

If the limit of the ratio of △z to △x exists when △x→0, then this limit value is called the partial derivative of the function z=f(x, y) with respect to x at (x0, y0), and is recorded as f'x(x0, y0) or. The partial derivative of function z=f(x, y) to x at (x0, y0) is actually the derivative of univariate function z=f(x, y0) at x0 after y is fixed at y0 as a constant.

Partial derivative in y direction

Similarly, if x is fixed at x0, let y have an increment △ Y, and if there is a limit, then this limit is called the partial derivative of the function z=(x, y) to y at (x0, y0). Write f'y(x0, y0).

People often say that the function y=f(x) is the relationship between the dependent variable and the independent variable, that is, the value of the dependent variable depends on only one independent variable, which is called a univariate function.

But in many practical problems, it is often necessary to study the relationship between the dependent variable and several independent variables, that is, the value of the dependent variable depends on several independent variables.

For example, the market demand of a commodity is not only related to its market price, but also related to the income of consumers and the prices of other substitutes of this commodity, that is, there are not one but many factors that determine the demand of this commodity. To study this kind of problems comprehensively, it is necessary to introduce the concept of multivariate function.

References:

Baidu encyclopedia-multivariate function