{"id":35237,"date":"2020-05-07T00:48:14","date_gmt":"2020-05-06T16:48:14","guid":{"rendered":"https:\/\/case.ntu.edu.tw\/blog\/?p=35237"},"modified":"2023-06-06T11:02:06","modified_gmt":"2023-06-06T03:02:06","slug":"%e8%bd%89%e8%a7%92%e9%81%87%e5%88%b0%e9%ac%bc-%e4%bb%a5%e6%9c%89%e9%99%90%e5%a4%a7%e9%a7%95%e9%a6%ad%e7%84%a1%e7%aa%ae%e5%b0%8f%e7%9a%84epsilon-delta-%e6%a0%b9%e8%99%9f-1%e7%9a%84%e6%95%85","status":"publish","type":"post","link":"https:\/\/case.ntu.edu.tw\/blog\/?p=35237","title":{"rendered":"\u3010\u79d1\u5b78\u53f2\u6c99\u9f8d\u3011\u3008\u8f49\u89d2\u9047\u5230\u9b3c\u2014\u2014\u4ee5\u6709\u9650\u5927\u99d5\u99ad\u7121\u7aae\u5c0f\u7684epsilon-delta\u3009&#038;\u3008\u6839\u865f-1\u7684\u6545\u4e8b\u3009"},"content":{"rendered":"<p style=\"text-align: center;\"><iframe src=\"https:\/\/www.youtube.com\/embed\/RM2mWtYJppk\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><span style=\"color: #993300;\"><strong>\u3008\u8f49\u89d2\u9047\u5230\u9b3c\u2014\u2014\u4ee5\u6709\u9650\u5927\u99d5\u99ad\u7121\u7aae\u5c0f\u7684epsilon-delta\u3009<\/strong><\/span><\/p>\n<p><span style=\"color: #800080;\"><strong>\u5fae\u7a4d\u5206\u767c\u5c55\u521d\u671f\uff0c\u4f7f\u7528\u76f4\u89ba\u5f0f\u7684\u8ad6\u8b49\uff0c\u96d6\u7136\u5be6\u52d9\u4e0a\u89e3\u6c7a\u4e86\u8a31\u591a\u554f\u984c\uff0c\u4f46\u662f\u4e5f\u9677\u5165\u908f\u8f2f\u4e0a\u7684\u56f0\u5883\uff0c\u8ad6\u8b49\u7684\u904e\u7a0b\u751a\u81f3\u88ab\u6279\u8a55\u662f\u300c\u770b\u5230\u9b3c\u300d\u3002\u6578\u5b78\u5bb6\u5f8c\u4f86\u4ee5\u300c epsilon-delta \u6975\u9650\u5b9a\u7fa9\u6cd5\u300d\uff0c\u89e3\u6c7a\u76f4\u89ba\u8ad6\u8b49\u7684\u908f\u8f2f\u554f\u984c\u3002\u672c\u8b1b\u6b21\u9664\u4e86\u4ecb\u7d39\u5176\u4f86\u9f8d\u53bb\u8108\u4e4b\u5916\uff0c\u4e5f\u8209\u51fa\u4eba\u985e\u70ba\u4e86\u7406\u89e3\u300c\u7121\u7aae\u300d\u9019\u500b\u6982\u5ff5\uff0c\u60f3\u51fa\u4f86\u7684\u8a31\u591a\u65b9\u6cd5\uff0c\u4e00\u8d77\u4f86\u8a8d\u8b58\u9019\u500b\u7f8e\u9e97\u6df1\u9083\u7684\u4e16\u754c\u3002<\/strong><\/span><\/p>\n<p><strong>\u8b1b\u5e2b\uff1a\u82f1\u5bb6\u9298\uff5c\u570b\u7acb\u81fa\u5317\u6559\u80b2\u5927\u5b78\u6578\u5b78\u66a8\u8cc7\u8a0a\u6559\u80b2\u5b78\u7cfb\u52a9\u7406\u6559\u6388<\/strong><\/p>\n<p>\u6bcf\u7576\u4eba\u985e\u6587\u660e\u5f9e\u90e8\u843d\u793e\u6703\u9032\u5165\u541b\u4e3b\u5236\u570b\u5bb6\u6642\uff0c\u5c0d\u65bc\u6578\u5b78\u7684\u9700\u6c42\u5c31\u6703\u76f8\u5c0d\u63d0\u9ad8\u3002\u8fb2\u696d\u570b\u5bb6\u4e00\u65e6\u767c\u5c55\u51fa\u6309\u7167\u571f\u5730\u5fb5\u7a05\u7684\u6982\u5ff5\uff0c\u4e08\u91cf\u7530\u5730\u9762\u7a4d\u5c31\u6703\u6210\u70ba\u91cd\u8981\u7684\u570b\u5bb6\u5de5\u4f5c\uff1b\u81f3\u65bc\u5efa\u9020\u5929\u6587\u53f0\u3001\u5e1d\u738b\u9675\u5be2\u3001\u5b97\u6559\u5efa\u7bc9\u7b49\u5de8\u5927\u5de5\u7a0b\uff0c\u5247\u9700\u8981\u8a08\u7b97\u5404\u7a2e\u4e0d\u540c\u985e\u578b\u7684\u9ad4\u7a4d\uff0c\u4ee5\u6b64\u4f30\u7b97\u6240\u9700\u7684\u4eba\u529b\u7269\u529b\u3002\u9019\u4e9b\u8a08\u7b97\u9762\u7a4d\u8207\u9ad4\u7a4d\u7684\u5177\u9ad4\u9700\u6c42\uff0c\u662f\u6578\u5b78\u767c\u5c55\u91cd\u8981\u7684\u52d5\u529b\u3002<\/p>\n<p>\u8a08\u7b97\u9762\u7a4d\u8207\u9ad4\u7a4d\u7684\u6578\u5b78\uff0c\u5177\u9ad4\u4f86\u8aaa\u662f\u7a4d\u5206\u7684\u7bc4\u7587\uff1b\u5c0f\u5b78\u6578\u5b78\u5c31\u6703\u6559\u6388\u7684\u4e09\u89d2\u5f62\u9762\u7a4d\u8207\u9577\u65b9\u9ad4\u9ad4\u7a4d\u7b97\u6cd5\uff0c\u53ef\u8aaa\u662f\u6700\u7c21\u55ae\u7684\u7a4d\u5206\u3002\u4f46\u662f\u8af8\u5982\u91d1\u5b57\u5854\u9019\u6a23\u7684\u9310\u9ad4\uff0c\u6216\u662f\u5713\u5f62\u4ee5\u53ca\u62cb\u7269\u7dda\u65cb\u8f49\u9ad4\u9019\u7a2e\u66f2\u7dda\u9ad4\uff0c\u7531\u65bc\u5b83\u5011\u4e0d\u80fd\u5206\u5272\u6210\u6709\u9650\u591a\u500b\u7c21\u55ae\u7684\u4e09\u89d2\u5f62\u6216\u9577\u65b9\u9ad4\uff0c\u56e0\u6b64\u5728\u8a08\u7b97\u5176\u9762\u7a4d\u6216\u9ad4\u7a4d\u6642\uff0c\u5c31\u5fc5\u9808\u7528\u4e0a\u4e00\u4e9b\u8ddf\u300c\u7121\u9650\u300d\u6709\u95dc\u7684\u601d\u8003\u65b9\u5f0f\u3002\u9019\u4e9b\u90fd\u662f\u73fe\u5be6\u751f\u6d3b\u4e2d\u6703\u78b0\u5230\u7684\u5be6\u969b\u554f\u984c\uff0c\u7136\u800c\u7121\u9650\u7684\u6982\u5ff5\u65e2\u4e0d\u7b26\u5408\u76f4\u89ba\uff0c\u5728\u908f\u8f2f\u4e0a\u4e5f\u5f88\u5bb9\u6613\u9020\u6210\u6096\u8ad6\uff0c\u56e0\u6b64\u53e4\u5e0c\u81d8\u4eba\u5728\u8ad6\u8b49\u6642\uff0c\u5c0d\u65bc\u7121\u9650\u7684\u6982\u5ff5\u7e3d\u662f\u907f\u4e4b\u552f\u6050\u4e0d\u53ca\u3002<\/p>\n<p>\u9019\u6a23\u5c0d\u65bc\u300c\u7121\u9650\u300d\u8af1\u83ab\u5982\u6df1\u7684\u614b\u5ea6\uff0c\u5230\u4e86\u79d1\u5b78\u9769\u547d\u7684\u6642\u4ee3\uff0c\u5fc5\u9808\u89e3\u6c7a\u5be6\u969b\u554f\u984c\u7684\u79d1\u5b78\u5bb6\u5011\uff0c\u958b\u59cb\u66f4\u52c7\u65bc\u4f7f\u7528\u7121\u9650\u7684\u6982\u5ff5\u52a0\u4ee5\u8ad6\u8b49\u3002\u6bd4\u65b9\u8aaa\u8cbb\u99ac\u8ddf\u7b1b\u5361\u5152\u8a0e\u8ad6\u66f2\u7dda\u7684\u5207\u7dda\u8ddf\u6cd5\u7dda\uff0c\u514b\u535c\u52d2\u8a08\u7b97\u6a62\u5713\u66f2\u7dda\u5167\u90e8\u7684\u9762\u7a4d\uff0c\u725b\u9813\u7814\u7a76\u884c\u661f\u904b\u52d5\u7684\u77ac\u9593\u901f\u5ea6\uff0c\u9019\u4e9b\u554f\u984c\u90fd\u5fc5\u9808\u7528\u4e0a\u300c\u6975\u9650\u300d\u7684\u601d\u8003\u3002\u725b\u9813\u8ddf\u840a\u5e03\u5c3c\u8332\u5728 17 \u4e16\u7d00\u767c\u5c55\u51fa\u4f86\u7684\u300c\u7121\u7aae\u5c0f\u91cf\u8a08\u7b97\u6cd5\u300d (infinitesimal calculus) \uff0c\u5f8c\u4f86\u6b78\u7d50\u767c\u5c55\u6210\u70ba\u5fae\u7a4d\u5206\u7684\u57fa\u790e\uff0c\u5728 18 \u5230 19 \u4e16\u7d00\u5ee3\u6cdb\u61c9\u7528\u65bc\u8a31\u591a\u79d1\u5b78\u8207\u5de5\u7a0b\u8a08\u7b97\uff0c\u89e3\u6c7a\u4e86\u8a31\u591a\u904e\u53bb\u7121\u6cd5\u89e3\u6c7a\u7684\u554f\u984c\u3002<\/p>\n<p>\u7136\u800c\u5fae\u7a4d\u5206\u9019\u500b\u65e2\u9055\u53cd\u4eba\u985e\u76f4\u89ba\uff0c\u53c8\u96e3\u4ee5\u8ad6\u8b49\u7684\u908f\u8f2f\u57fa\u790e\uff0c\u4ecd\u7136\u96e3\u4ee5\u8aaa\u670d\u8a31\u591a\u54f2\u5b78\u5bb6\u8207\u6578\u5b78\u5bb6\u3002\u5c0d\u65bc\u6301\u7591\u8005\u4f86\u8aaa\uff0c\u9019\u500b\u65e2\u4e0d\u662f\u6709\u9650\u91cf\uff0c\u4e5f\u4e0d\u662f\u7121\u9650\u5c0f\u91cf\uff0c\u53c8\u4e0d\u662f\u7a7a\u7121\u7684\u300c\u6f38\u6f38\u6d88\u5931\u7684\u589e\u91cf\u300d\uff0c\u7c21\u76f4\u50cf\u662f\u300c\u8f49\u89d2\u9047\u5230\u9b3c\u300d\u4e00\u6a23\u96e3\u7e8f\u3002\u9019\u500b\u96e3\u984c\u5728 19 \u4e16\u7d00\u6578\u5b78\u5bb6\u67ef\u897f (Augustin-Louis Cauchy) \u3001\u5e03\u723e\u624e\u8afe (Bernhard Bolzano) \u3001\u9b4f\u723e\u65af\u7279\u62c9\u65af (Karl Weierstrass) \u7b49\u4eba\u7684\u52aa\u529b\u4e4b\u4e0b\uff0c\u767c\u5c55\u51fa epsilon-delta \u6975\u9650\u5b9a\u7fa9\uff0c\u8b93\u6574\u500b\u8ad6\u8b49\u904e\u7a0b\u5168\u90fd\u662f\u6709\u9650\u91cf\u7684\u56db\u5247\u904b\u7b97\u8ddf\u958b\u65b9\uff0c\u5b8c\u5168\u907f\u958b\u7121\u7aae\u5c0f\u91cf\u7684\u9ebb\u7169\u8655\uff0c\u56e0\u6b64\u6210\u70ba\u73fe\u4ee3\u5fae\u7a4d\u5206\u7406\u8ad6\u7684\u57fa\u77f3\u3002<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #993300;\"><strong>\u3008\u6839\u865f-1\u7684\u6545\u4e8b\u3009<\/strong><\/span><\/p>\n<p><span style=\"color: #800080;\"><strong>\u4ee5\u8907\u6578\u70ba\u57fa\u790e\u7684\u8907\u8b8a\u51fd\u6578\uff0c\u53ef\u61c9\u7528\u65bc\u6d41\u9ad4\u529b\u5b78\u3001\u71b1\u529b\u5b78\u3001\u96fb\u78c1\u5b78\u7b49\u9818\u57df\uff0c\u662f\u5be6\u7528\u6027\u76f8\u7576\u9ad8\u7684\u6578\u5b78\u601d\u7dad\u3002\u7136\u800c\u6839\u865f -1 \u9019\u500b\u9055\u53cd\u4eba\u5011\u76f4\u89ba\u7684\u6982\u5ff5\uff0c\u76f4\u5230 19 \u4e16\u7d00\u624d\u6b63\u5f0f\u70ba\u6578\u5b78\u754c\u6240\u63a5\u7d0d\u3002\u672c\u8b1b\u6b21\u4ecb\u7d39\u4ee5\u6839\u865f -1 \u70ba\u4ee3\u8868\uff0c\u865b\u6578\u8207\u8907\u6578\u6982\u5ff5\u7684\u767c\u5c55\u6b77\u53f2\u8108\u7d61\u3002<\/strong><\/span><\/p>\n<p><strong>\u8b1b\u5e2b\uff1a\u8607\u610f\u96ef\uff5c\u81fa\u5317\u5e02\u7acb\u5927\u5b78\u6578\u5b78\u7cfb\u526f\u6559\u6388<\/strong><\/p>\n<p>\u5728\u570b\u4e2d\u7684\u6578\u5b78\u6559\u5b78\u4e2d\uff0c\u6839\u865f -1 \u7684\u7b54\u6848\u662f\u7121\u89e3\uff1b\u5230\u4e86\u9ad8\u4e2d\u4ee3\u6578\uff0c\u5f15\u9032\u865b\u6578 i \u7684\u6982\u5ff5\u4e4b\u5f8c\uff0c\u6839\u865f -1 \u8b8a\u6210\u6709\u89e3\u3002\u6211\u5011\u5be6\u969b\u4e0a\u8a62\u554f\u9ad8\u4e2d\u540c\u5b78\u5c0d\u65bc\u865b\u6578\u7684\u770b\u6cd5\uff0c\u4ed6\u5011\u591a\u534a\u89ba\u5f97\u865b\u6578\u7684\u6982\u5ff5\u865b\u5e7b\u4e0d\u5be6\uff0c\u4e0d\u77e5\u9053\u9019\u500b\u300c\u4e0d\u5b58\u5728\u300d\u7684\u6578\u6709\u4ec0\u9ebc\u5be6\u969b\u7528\u9014\uff0c\u4f3c\u4e4e\u53ea\u662f\u6578\u5b78\u5bb6\u70ba\u4e86\u65b9\u7a0b\u5f0f\u4e00\u5b9a\u8981\u6709\u89e3\uff0c\u786c\u662f\u5275\u9020\u51fa\u4f86\u81ea\u5713\u5176\u8aaa\u7684\u6771\u897f\u3002<\/p>\n<p>\u9019\u6a23\u7684\u56f0\u60d1\u4e0d\u662f\u53ea\u6709\u9ad8\u4e2d\u540c\u5b78\u624d\u6709\u3002 12 \u4e16\u7d00\u5370\u5ea6\u5091\u51fa\u7684\u6578\u5b78\u5bb6\u5a46\u4ec0\u8fe6\u7f85 (Bh\u0101skara II) \uff0c\u5c31\u4e0d\u76f8\u4fe1\u6709\u81ea\u4e58\u7b49\u65bc -1 \u9019\u7a2e\u6578\u5b58\u5728\uff0c\u4e5f\u4e0d\u9f13\u52f5\u4eba\u5011\u53bb\u9032\u884c\u300c\u4e0d\u5207\u5be6\u969b\u300d\u7684\u601d\u8003\u3002 16 \u4e16\u7d00\u7fa9\u5927\u5229\u6578\u5b78\u5bb6\u5361\u4e39\u8afe (Girolamo Cardano) \uff0c\u6709\u4e00\u6b21\u5229\u7528\u8ca0\u6578\u958b\u6839\u865f\u9032\u884c\u904b\u7b97\u6642\uff0c\u4e5f\u89ba\u5f97\u826f\u5fc3\u4e0d\u5b89\uff0c\u8868\u793a\u7b97\u6578\u7684\u767c\u5c55\u300c\u96d6\u7136\u7cbe\u7dfb\uff0c\u537b\u4e0d\u4e2d\u7528\u300d\u3002\u7b1b\u5361\u5152\u3001\u725b\u9813\u8ddf\u840a\u5e03\u5c3c\u8332\uff0c\u66f4\u662f\u8a8d\u70ba\u8ca0\u6578\u7684\u5e73\u65b9\u6839\u662f\u865b\u5e7b\u7684\uff0c\u6c92\u6709\u5be6\u8cea\u610f\u7fa9\uff0c\u7406\u6027\u4e0a\u4e0d\u80fd\u63a5\u53d7\u3002<\/p>\n<p>\u7b2c\u4e00\u4f4d\u70ba\u865b\u6578\u8aaa\u8a71\u7684\uff0c\u7576\u6578 16 \u4e16\u7d00\u7fa9\u5927\u5229\u4ee3\u6578\u5b78\u5bb6\u90a6\u8c9d\u529b (Rafael Bombelli) \uff0c\u4ed6\u5728\u50b3\u4e16\u8457\u4f5c\u300a\u4ee3\u6578\u5b78\u300b\u8a0e\u8ad6\u8ca0\u6578\u7684\u5e73\u65b9\u6839\uff0c\u5229\u7528\u5361\u4e39\u8afe\u767c\u8868\u7684\u4e09\u6b21\u65b9\u7a0b\u5f0f\u89e3\uff0c\u767c\u5c55\u51fa\u6839\u865f -1 \u7684\u904b\u7b97\u6cd5\u5247\uff0c\u5960\u5b9a\u4e86\u865b\u6578\u7406\u8ad6\u7684\u57fa\u77f3\u3002 17 \u4e16\u7d00\u6cd5\u570b\u6578\u5b78\u5bb6\u57fa\u62c9\u5fb7 (Albert Girard) \uff0c\u5247\u8aaa\u6211\u5011\u53ef\u4ee5\u85c9\u8457\u9019\u4e9b\u4e0d\u53ef\u80fd\u7684\u89e3\uff0c\u80af\u5b9a\u4e00\u822c\u4ee3\u6578\u6cd5\u5247\uff0c\u56e0\u6b64\u6709\u5176\u61c9\u7528\u50f9\u503c\uff0c\u4e0d\u7136\u9664\u6b64\u4e4b\u5916\u4e5f\u5225\u7121\u4ed6\u89e3\u3002 18 \u4e16\u7d00\u5049\u5927\u7684\u6578\u5b78\u5bb6\u5c24\u62c9\uff0c\u4e5f\u5728\u4ed6\u91cd\u8981\u7684\u4ee3\u6578\u8457\u4f5c\u300a\u4ee3\u6578\u6307\u5357\u300b\u4e2d\uff0c\u8aaa\u4e86\u300c\u865b\u6578 i \u65e2\u4e0d\u662f 0 \uff0c\u4e5f\u4e0d\u5927\u65bc 0 \u6216\u5c0f\u65bc 0 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