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 important book, Paradoxien des Unendlichen. One of his results, now known as the theorem of Bolzano, was first published in 1817.There are several important theorems of calculus in which continuity plays a centralrole. In each of these theorems, continuity of a real-valued function on a closed interval ofthe real line is an essential hypothesis；that is, if the function is not continuous on aclosed interval, then none of the statements below is necessarily true. If fis defined on aclosed interval [a, 6]» then continuity at the left-hand endpoint, a,means that linii-fl* /(x)= /(a) * and continuity of fat 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 fattains an absolute minimum value, m, at some point c£[a, b}* and anabsolute maximum value. M, at some pointb\.