拉格朗日中值定理是微分学的基础定理之一,很多时候我们可以借助其研究导数的性质,从而加深对函数在整个定义域区间上的整体认识.本文讨论定理在证明函数一致连续性、有界性和单调性时的应用。因为拉格朗日中值定理沟通了函数与其导数的联系,很多时候我们可以借助其导数,研究函数的性质,从而加深对函数在整个定义域区间上的整体认识.比如研究函数在区间上的一致连续性,单调性,有界性等,都可能用到拉格朗日中值定理的结论.通过对函数局部性质的研究把握整体性质,这是数学研究中一种重要的方法。一、证明函数一致连续性例1证明若函数f(x)于有穷或无穷的区间(a,b)内有有界的导 Lagrange’s median theorem is one of the basic theorems of differential theory, and many times we can use it to study the properties of derivatives, so as to deepen the overall understanding of the function over the entire domain of definition. This paper discusses the theorem in proving the consistent function , Boundedness and monotonicity of the application. Because the Lagrange median theorem communicates the connection of a function with its derivative, we often use it’s derivatives to study the properties of the function, thereby deepening the overall understanding of the function over the entire domain of the definition. For example, Uniform continuity, monotonicity, boundedness and so on, may conclude that Lagrange’s median theorem can be used.According to the study of the local properties of the function, the overall property is grasped, which is an important method in mathematical research. First, the proof function is consistent with continuity Example 1 proves that if the function f (x) in a finite or infinite range (a, b) has a bounded guide