\u6570\u5b66\u4e13\u4e1a\u82f1\u8bed_119 - \u6570\u5b66\u4e13\u4e1a\u82f1\u8bed recognize that many\"obvious statements about continuous functions require proof His observations

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数学专业英语 recognize that many "obvious" statements about continuous functions require proof. His observations concerning continuity were published posthumously in 1850 in an im portant book, Paradoxien des Unendlichen. One of his results, now known as the theo rem of Bolzano, was first published in 1817. There are several important theorems of calculus in which continuity plays a central role. In each of these theorems, continuity of a real-valued function on a closed interval of the real line is an essential hypothesis that is, if the function is not continuous on a closed interval, then none of the statements below is necessarily true. If f is defined on a closed interval [a, 6]» then continuity at the left-hand endpoint, a, means that linii-fl* /(x) = /(a) * and continuity of f at the right-hand endpoint, b, means that lim—- / ( x)=/(6). The Extreme Value Theorem : If / is a function that's continuous on a closed interval [a, b], then f attains an absolute minimum value, m , at some point c£[a, b} * and an absolute maximum value. M, at some point b\.

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