在一点有定义,和在一点连续有区别吗 函数在一点上没有定义,那么函数在这一点上一定不连续吗
\u5728\u4e00\u70b9\u6709\u5b9a\u4e49\uff0c\u548c\u5728\u4e00\u70b9\u8fde\u7eed\u6709\u533a\u522b\u5417\u4ec0\u4e48\u662f\u5b9a\u4e49\u4e0d\u4e00\u6837\u554a
\u4e3e\u4e2a\u4f8b\u5b50
f\uff08x\uff09=-1\uff0cx0
\u6709\u5b9a\u4e49 \u5355\u4e0d\u8fde\u7eed
\u9996\u5148\uff0c\u8fde\u7eed\u7684\u5b9a\u4e49\u662ff\uff08x\uff09\u5728x=x0\u70b9\u5904\u7684\u6781\u9650\u503c\u7b49\u4e8e\u51fd\u6570\u503c\u3002
\u6240\u4ee5\u4ece\u5b9a\u4e49\u5c31\u53ef\u4ee5\u770b\u5230\uff0c\u5982\u679cf\uff08x\uff09\u5728x=x0\u70b9\u5904\u90fd\u6ca1\u5b9a\u4e49\u7684\u8bdd\uff0c\u5c31\u4e0d\u53ef\u80fd\u6709\u51fd\u6570\u503c\uff0c\u5f53\u7136\u5c31\u4e0d\u53ef\u80fd\u6ee1\u8db3\u6781\u9650\u503c\u7b49\u4e8e\u51fd\u6570\u503c\u7684\u8981\u6c42\uff0c\u5c31\u4e0d\u53ef\u80fd\u8fde\u7eed\u3002
\u81f3\u4e8e\u4f60\u8bf4\u7684\u201c\u8ba8\u8bba\u51fd\u6570f(x)=x^2sin1/x (x\u22600) 0 (x=0)\u5728\u70b9x=0\u5904\u7684\u8fde\u7eed\u6027\u4e0e\u53ef\u5bfc\u6027
\u4f46\u662f\u50cf\u8fd9\u9053\u9898\uff0cx\u57280\u9664\u6ca1\u6709\u5b9a\u4e49\uff0c\u90a3\u8fd8\u4e3a\u4ec0\u4e48\u8981\u8ba8\u8bba\u5728x=0\u5904\u7684\u8fde\u7eed\u6027\u548c\u53ef\u5bfc\u6027\u5462\u201d
\u662f\u4f60\u7406\u89e3\u9519\u8bef\u3002
\u8fd9\u662f\u4e2a\u5206\u6bb5\u51fd\u6570\uff0c\u5728x\u22600\u7684\u65f6\u5019\uff0c\u51fd\u6570\u5f0f\u662fx^2sin1/x (x\u22600)\uff0c\u800c\u5728x=0\u7684\u65f6\u5019\uff0c\u4eba\u4e3a\u7684\u628a\u51fd\u6570\u503c\u5b9a\u4e49\u4e3a0\uff080 (x=0)\uff09
\u6240\u4ee5\u8fd9\u4e2a\u51fd\u6570\u5728x=0\u70b9\u5904\u662f\u6709\u5b9a\u4e49\u7684\uff0c\u5b9a\u4e49\u7684\u51fd\u6570\u503c\u5c31\u662ff\uff080\uff09=0\u3002
\u5176\u5b9e\u5c31\u662f\u539f\u672c\u51fd\u6570\u5f0fx^2sin1/x\u867d\u7136\u5728x=0\u70b9\u5904\u65e0\u5b9a\u4e49\uff0c\u4f46\u662f\u5f53x\u8d8b\u8fd1\u4e8e0\u7684\u65f6\u5019\uff0c\u6709\u6781\u9650\uff0c\u6781\u9650=0\uff0c\u6240\u4ee5\u4eba\u4e3a\u7684\u8865\u5145x=0\u5904\u7684\u51fd\u6570\u503c\u4e3af\uff080\uff09=0\u7684\u8bdd\uff0c\u5c31\u628a\u539f\u6765\u4e0d\u8fde\u7eed\u7684\u51fd\u6570\u5316\u4e3a\u8fde\u7eed\u7684\u51fd\u6570\u4e86\u3002\u6240\u4ee5x=0\u8fd9\u7c7b\u70b9\u88ab\u79f0\u4e3ax^2sin1/x\u7684\u53ef\u53bb\u95f4\u65ad\u70b9\uff0c\u867d\u7136\u662f\u95f4\u65ad\u70b9\uff0c\u4f46\u662f\u53ef\u4ee5\u4eba\u4e3a\u6539\u53d8\u95f4\u65ad\u70b9\u5904\u51fd\u6570\u503c\u7684\u5b9a\u4e49\u6765\u5316\u4e3a\u8fde\u7eed\u51fd\u6570\u3002
\u4f60\u5bf9\u5206\u6bb5\u51fd\u6570\u7684\u7406\u89e3\u4e0d\u5230\u4f4d\u554a\u3002
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