①Give a mapping f:X→Y,if for every x1,x2∈X with satisfying x1≠x2, we have f(x1)≠f(x2) Then f is called an injective
②Give a mapping f:X→Y,if for every y∈Y,there is an x∈X, such that f(x)=y ,then f is called a surjective x:domain(定义域) range(值域) Y(Ranf/Rancf) Preimage(原像)→image(像)
③XR,YR then f X→Y,is called a function and is denoted by(被…表示)y=f(x) properties of function(函数的特性)
1 boundedress有界性 boundary边界 interval区间 bounded有界的
Give a fuction y=f(x) with x∈D if for every x in D,there exists an M＞0 satisfying 丨f(x)丨≤M
Then we called f(x) is bounded on D
2 monotone(单调性) Given a function y=f(x) with X∈I if for every x1,x2 in I with satisfying x1 less than x2,we have f(x1) less than or equal to f（x2），then we called f（x）is a monotone increasing function on I
3 add/even function 奇（偶）函数
表示关于原点对称be symmetric -----origin （origin of symmetry）
4The necessary and sufficient condition of seqvence xn being convergent（收敛）is that ∀ξ＞0，∃N，such that for arbitrary n，m＞N，the inequality丨f（x）-A丨＜ holds
{xn}收敛<=>ξ＞0，∃N使得当n，m＞N有丨xA-xm丨＜ξI

A large variety of scientific problems arise in which one tries to determine something from its rate of change. For example, we could try to compute the position of a moving particle from a knowledge of its velocity or acceleration. Or a radioactive substance may be disintegrating at a known rate and we may be required to determine the amount of material present after a given time. In examples like these, we are trying to determine an unknown function from prescribed information expressed in the form of an equation involving at least one of the derivatives of the unknown function. These equations are called differential equation, and their study forms one of the most challenging branches of mathematics.
面对大量出现的各种科学问题，人们会试图根据事物的变化率来确定。例如，我们会根据一个运动粒子的速度或加速度来确定它的位置；或者某种放射性物质以一定的速率裂变，我们可能要确定一段时间内放射出的物质量。诸如此类，我们试图通过已知信息确定未知功能，并通过一定的公式表达出来，公式中包含这种未知功能的至少一种变量。这种公式我们称之为微分公式，是数学领域最具挑战性的分支学科之一。I