由 linch6123 於 星期四 二月 23, 2006 10:05 am
#ed_op#DIV#ed_cl#Let f(x)=sin(1/x) for x in (0,1)#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#then#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# f(1/(2npi+pi/2))-f(1/(2npi))=1#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#but#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# ( 1/2npi+pi/2)-(1/2npi)→ 0 when n is large #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#therefore #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#f(x)=sin(1/x) for x in (0,1) is continuous and bounded ,but f(x) is not uniformly continuous #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl#