Necessary conditions for differentiability of functions: If a function is differentiable at a certain point, it must be continuous at that point; If a binary function is differentiable at a certain point, then the partial derivatives of the function to x and y must exist at that point.

Sufficient conditions for the function to be differentiable: If the partial derivatives of the function to x and y exist in a neighborhood of this point and are continuous at this point, then the function is differentiable at this point.

The differentiable condition of multivariate function is that both partial derivatives of f(x, y) exist at point (x0, y0).

Knowledge expansion:

Function is a relationship that expresses the dependence between two or more variables in mathematics. This function maps input values to output values. No matter how the input value changes, the output value will change according to the definition of the function.

The definition of a function can be a mathematical expression, an algorithm, a table or any other form. The input of a function is usually called an independent variable, and the output of a function is called a dependent variable. The relationship between independent variables and dependent variables is functional and can be expressed by an equation.

Functions can be classified in different ways. For example, according to the nature of functions, they can be divided into linear functions, quadratic functions, power functions, trigonometric functions and so on. According to the purpose of the function, it can be divided into calculation function, probability function, statistical function and so on. Variables can be divided into univariate functions and multivariate functions according to functions.

In mathematics, some commonly used functions include:

1, linear function: a linear function is a power function, and its image is like a straight line. The formula of linear function is y=kx+ b, where k and b are constants.

2. Quadratic function: Quadratic function is a quadratic function, which is like a parabola. The formula of quadratic function is y = ax 2+bx+c, where a and bc are constants.

3. Power function: Power function is exponential function, and its image is curve. The formula of power function is y = x n, where n is a constant.

4. Calculation function: Calculation function refers to functions used for various mathematical calculations, such as absolute value function, square root function, logarithmic function, etc.

In practical application, functions can be used to solve various problems, such as optimization problems, prediction problems, calculation problems and so on. Functions are widely used in mathematics, physics, engineering, computer science and other fields.