Question

证明：函数f(x)在（a,b)内有界的充分必要条件是f(x)在(a,b)内既有上界,又有下界.

Answer 1

1.若f(x)在(a,b)内有界,则存在M,恒有 |f(x)|≤M,即-M≤f(x)≤M,所以f(x)在有上界M,下界-M

2.若f(x)在有上界M,下界N,则恒有N≤f(x)≤M,设T=Max{ |M|,|N| },则恒有-T≤N≤f(x)≤M≤T,

即|f(x)|