函数的零点的充要条件以及该条件下零点跟定义域的连续条件的证明

The Necessary and Sufficient Conditions for the Zero Point of a Function and the Proof of the Continuity Condition between the Zero Point and the Domain under This Condition

摘要: 在中学时我们曾经学过零点存在定理，当我们随着做题以及学习的深入，我们会发现一些函数似乎并不能采用零点存在定理来说明零点存在。例如在连续的函数中我们有函数$f\left(x\right)=x^2$ 或者$y=\left|x\right|$ 还有 $f\left(x\right)=\left[x\right]\left(x\ge 0\right)$ 等等，或者说一些间断的函数比如黎曼函数$R\left(X\right)$ ，在$R\left(X\right)$ 中，当$x=0$ 或者$x=1$ 时取到零点，但是我们会发现，因为在定义域(0, 1)取的是无理数，所以，该函数的零点，我们找不到零点存在定理的条件，但又找得到零点。所以综上所诉，我们总能发现$f\left(x\right)=0$ 这个命题似乎跟零点存在定理的关系是满足零点存在定理一定是零点，但零点不一定满足零点存在定理。那有没有跟任何函数的零点存在时，会有一个命题会与之同时存在。或者说$f\left(x\right)=0$ 的充分必要条件的命题存在。带着这个问题，我们来进行函数零点的讨论。

Abstract: In high school, we learned about the zero existence theorem. As we delve deeper into solving problems and learning, we may find that some functions cannot be explained by the zero existence theorem. For example, in continuous functions, we have functions $f\left(x\right)=x^2$ or $y=\left|x\right|$ and $f\left(x\right)=\left[x\right]\left(x\ge 0\right)$ , or in some discontinuous functions such as the Riemann function $R\left(X\right)$ , in $R\left(X\right)$ , when $x=0$ or $x=1$ , the zero point is taken. However, we will find that because we take irrational numbers in the domain, we cannot find the conditions for the existence theorem of the zero point of the function, but we can find the zero point. So, based on the above statements, we can always find that the proposition $f\left(x\right)=0$ seems to be related to the existence theorem of zeros, that is, satisfying the existence theorem of zeros must be zeros, but zeros do not necessarily satisfy the existence theorem of zeros. When there is a zero point with any function, there will be a proposition that exists simultaneously with it. Or in other words, the proposition of sufficient and necessary conditions for $f\left(x\right)=0$ exists. With this question in mind, let’s discuss the function zeros.