以下內容來自中科大數學分析教程P73,定理2.4.7
\(函數在x_{0}點的極限的定義\)
\(若存在l，\forall \epsilon>0，\exists\delta>0，使得當|x-x_{0}|<\delta\)
\(則有|f(x)-l|<\epsilon,即稱l為f(x)當x趨近於x_{0}的極限\)
\(定理：函數f(x)在x_{0}處有極限的充要條件是\forall \epsilon>0,\exists\delta>0,\)
\(\quad\quad 使得任意x_{1},x_{2}\in U(x_{0},\delta)時，有\)
\(\quad\quad |f(x_{1})-f(x_{2})|<\epsilon\)
證明：
1.必要性
\(若f(x)在x_{0}點的極限為l，即\forall \frac{\epsilon}{2}>0,\exists\delta,當x_{1},x_{2}\in U(x_{0},\delta)\)
\(有|f(x_{1})-l|<\frac{\epsilon}{2},|f(x_{2})-l|<\frac{\epsilon}{2}\)
\(則：|f(x_{1})-f(x_{2})|=|f(x_{1})+l-l-f(x_{2})|\)
\(\quad\quad \leqslant |f(x_{1})-l|+|f(x_{2})-l|\)
\(\quad\quad\leqslant\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon\)