Firstly, starting with discrete series, we define the limit, convergence or divergence of series, the properties of convergence sequence, convergence criteria and so on.
Equations with unknowns are equations, and mathematics was first developed in counting. The combination of addition, subtraction, multiplication, division and idempotency between numbers and unknowns forms algebraic equations: one-dimensional linear equations, one-dimensional quadratic equations, two-dimensional linear equations and so on. However, with the emergence of the concept of function and the introduction of differential and integral operation based on function, the category of equation is wider, and the unknowns can be mathematical objects such as functions and vectors, and the operation is no longer limited to addition, subtraction, multiplication and division.
Then discuss the limit of function, start with the definition, transfer the idea of the limit of sequence, discuss the nature of the limit of function, and connect the sequence with the function through Heine principle; Related knowledge points, such as property theorems, can be learned by analogy with sequence. After all, series is a discrete quantity (series can be understood as a function of independent variables and natural numbers), and functions are mainly continuous quantities.
Because mathematics has changed from constant mathematics to variable mathematics, the contents of equations are enriched, because mathematics has introduced more concepts and more operations, thus forming more equations. The development of other natural sciences, especially physics, also directly puts forward the demand of solving equations and provides a large number of research topics.
Because the domain of continuous function is a set of real numbers, the sequence can be regarded as a function defined on a set of positive integers. The function introduces continuity and uniform continuity defined by limits, and then gives some property theorems of continuous function, such as boundedness, zero or median, maximum and so on.
Differential equation refers to an equation containing unknown functions and their derivatives. The unknowns of this kind of equations are functions. Different from functional equation, it has derivative operation on unknown function, which can be high-order derivative. But if the unknown function in the equation contains only one independent variable, then the differential equation is an ordinary differential equation.
In order to further study the properties of function, the derivative and differential of function are defined by limit, and the monotonicity, extreme value or maximum value, concavity and convexity of function are discussed by introducing derivative law and differential mean value theorem, and the relationship between derivability and continuity of function is also discussed.