1、How did Dirichlet make a discontinuity with continuityOn the continuityLife on MathematicsA story of the square tableSails cruise back along the Yangtze River, thousands of miles;West wind blow across the Yang Pass, tens of times.The scene of the mortal world fills my eyes.Standing by the fences,I s
2、ee newborn wild geese hover facing the sunset.Over the Chu State clouds are drifting across the sky;The Xiang River keeps flowing eastward and riding high.Life is full of parting and grief, I dont know why.Against the evening glow,A few swans past the tower gracefully fly.A story of the square table
3、Every day we see dynamic and continuous scenes of nature. Continuity isour most basic impression of the world around us since we were born,and as a result, everything continues to grow. We dont have to worryabout the sudden interruption of oxygen supply when walking. Theproud plum blossoms will not
4、stop opening because of the cold wind andrain. The flowing Yangtze River is even more unlikely to stand still by theRed Cliff for a second at one evening to recall the ancient battlefield.The Story of the Square TableLet the table not shake on the ground.To find an appropriate angle?A mathematical m
5、odel: the square table problem.Model buildingThus, the square table problem can be attributed to a mathematical problem:()f = 0 t able f oo t A and C t ouc h t he gr o un d ()g = 0 t abl e f oot B and D t ouc h t he gr o un d xB AACDDBCIntermediate value theoremof continuous functions: the sum of th
6、e distance from the table foot A to the ground and foot C: the sum of the distance from the table foot B to the ground and foot D According to life experience, at least square table can stand on three feet at any time, which means for any , either or is always zero. ( )f ( )g( )f ( )gFor and , ( )f
7、( )g ( ) ( )fg=0there is . If set, prove that there must be an( ) , ( )gf=0 0 0 0angle to make ( ) ()=00 00Proof of the conclusionProof: (1) If , then can prove the conclusion.(2) If , We construct a functionIts easy to haveNow rotate the square table 90, then the diagonal of the square table is int
8、erchanged, the state of the square table does not change. Thus, there is ( )f =00 =0 0( )f ( ) ( ) ( )h f g =( ) ( ) ( )h f g= 0 0 0 0( )fg= 002 ( )gf =2h 0Obviously, is continuous functions,by the intermediate value theorem forcontinuous functions, there must be a , 0 0 2to make( )h 0 0which means
9、( ) ( )fg=00Then ( ) ( )=0( ) ( )=0The original problem was solved.( )hQuestionw he n ( 0 ) 0h ， ( ) 02h ， c an you ded uce t he e xiste nce of 0 (0 , )2 ， s o t hat 0( ) 0h = ？ He Youmeis question:ConsiderO xy1-11101( ) 02102xf x xx =0 ( 1,1) 0( ) 0f =What make ?The brother said: it is not a contin
10、uous function!Does the continuity make a discontinuityWhat kind of definition does continuity have so that it cansuccessfully blend in dynamic and static states? How didDirichlet make a discontinuity with continuity? Why iscontinuity and discontinuity distinguished clearly at themoment when the teac
11、her steps on the podium?The Meaning of Continuity 0 0 1 2 0 The curve is not broken BrokenAll continuous function can be seen? Dirichlet function 2( ) = l im ( l im ( c os( ! ) ) )jkjf x k x The formula of the Dirichlet function with two limits can be expressed as 1() 0 whe n x is a rationa l num be
12、 rfx whe n x is an irrationa l num be r= The Dirichlet function in piecewise form can be written ask and j are natural numbersCan you draw it?1()0whe n x is a rationa l num be rfxwhe n x is an irrationa l num be r= The Dirichlet function in piecewise form isContinuity of the Dirichlet function Defin
13、ite The function is continuous at a iflim = Otherwise, the function is discontinuity.Definite For every 0 there is some 0 such that, for all x, if 0 ,then 0 xf ( x )O1x 0 - x0 + x 0 1Ra t i ona l num be rIrr a t i on a l n um be rx Just take a rational number point to prove it. 0 = 1. Take 0, Theres
14、 always a irrational numberpoint x in the neighbourhood 0 So, the rational number point 0 is not the continuous point of the Dirichlet function.Continuity of the Dirichlet function 1()0whe n x is a rationa l num be rfxwhe n x is an irrationa l num be r= 0 0 210 0lim ( ) ( )xxf x f x =A Practical Met
15、hod for Determining Continuityprocess of approximation :from the left or the right Whether the function changes “continuously” or “jumps”? Walking onto the podiumThe continuity all is about one point and a collective characteristicWhy is continuity all about one point? How does it make theproperties
16、 of a point into a collective characteristic? Why isthe memory of an era often marked by an event, so thatcontinuity and discontinuity is cleverly connected?f(x0) 0 f(x0) 0The Benefits of Continuity? f(x0) exists f(x) is bounded?Ice will melt at the temperature above 0F o r an y 0 , th e r e e x is
17、ts 0 so th a t w h e n x is in t h i s n e ig h b o r h o o d o f 0 x , ( ) ( )0 0f x f x . ( ) ( ) ( )00f f x f x +Local good propertiesTheorem 1： If is continuous at the point , then is boundedin a neighborhood of the point .()fx 0 x()fx0 xTheorem 2： If is continuous atthe point , and if ,then the
18、re must be a certainneighborhood of , and in thisdomain .()fx0 x 0 x ( ) 0( 0)f x or( ) 0( 0)f x orProperties of neighborhood : collective feature of a function near a point.Traffic jams caused by traffic accidentsLocal good propertiesFlow of people getting off the train Drinking teaThe Continuity o
19、f Chinese HistoryAustralian colleagues told me thatChinese history is continuous.We are curious about ourselves. Chinese Historiographer Cang JieThe privilege of not being restricted by the emperors.Unfortunately, many civilizations in the world disappeared.100 cultural relics Showcasing 2 million y
20、ears of human historyIn 2016, the British Museum held exhibitions around the world, including in Beijing and Shanghai.In 2010, the British Museum did a sensation in the world. Under the lead of NeilMacGregor, then Director of the British Museum, the 100 most representative itemsin the museum were se
21、lected and the museum made a 20-week program at the BBCBroadcasting Corporation. This show featured 100 cultural relics to tell about humancivilization, from the Stone Age to 2010.In 2011, the show set a record of having 11 million online podcast listeners, and it was also edited by Penguin Press in
22、to a book A History of the World in 100 ObjectsContinuity and discontinuityA film of Ang Lee director, Life of Pi. ( ) 0 , 1 =u x x xnnnux n n nnn= 11 0, )2 1 2 , )()11 , 1 1( ) ( )nu x u x nl im ( ) ( )nn u x u x + =The linear functionselectionSet It is said to be “continuous records”, but the hist
23、oriographers cannotrecord ancient events minute by minute. They just recorded some majorevents and used some intermittent events to describe a continuous process.What we feel is that the historical record is continuous.Measurement of Your HeightIntermittent records confirm the continuous growth. We
24、humans filter out constantly repetitive events, just remember intermittent facts.An awkward situation of the Loch Ness monsterWhy does continuity achieve a holistic declarationthrough the intermediate value theorem? What kind ofmathematical principle does the story “Bamiao Zhuzhang”(to pull up the s
25、eedlings to help them grow) reflect? Howcan continuity bring the Loch Ness monster into anawkward situation? What kind of mathematical formdoes hun and po adhere to?A Lesson from “Bamiao Zhuzhang”OA motto in classical physics, “Everything is continuous in nature.” The growth of the seedlings require
26、s continuity.Pull up the seedlings to help them growI f the f u nction f (x ) is co nti nuo u s in a cl osed int er v al a, b, and f (a) and f (b) h a v e opp osit e sig ns, the n w e can f ind a p oint in the op en in t er v al ( a, b) t o m ak e . Temperature changes continuouslyI f is c ontinuous
27、 on , a nd ( ) ( )or ( ) ( ) , the n the r e is som e in , suc h tha t ( ) .T he ore m f a b f a c f bf a c f b x a b f x c =Intermediate value theorem The intermediate value theorem highlights thecompleteness of a continuous function over itsrange, and the function curve must pass througheach point
28、 in the range.Can you make a tree grow from asapling suddenly to a toweringtree? This is impossible.Loch Ness Water MonsterBiological chain.The photo of the MonsterMany investigations were adopts. A continuous biological chain was not found in the lake.“Water still flows, though we cut it with our s
29、words, and sorrows return, though we drown them with wine”“Water still flows, though we cut it with our swords”. This versedescribes that one hopes to achieve something but cannot have a result.The development of emotions cannot be interrupted abruptly, nor can itbe cleared instantly.“Forget it” or
30、“Dont miss him(her)” Actually, our comforting wordswont work.Through pouring out slowly, yourfriends hurt feelings will graduallybe relieved.“Hun” means god of yang(positive, male)“Hun” and “Po”“Po” means god of yin (negative, female)“Hun” controls peoples consciousness when they are awake; “Po” con
31、trols over people when they are asleep. Wake up before the alarm goes off. Can you open one eye and close one while sleeping？Some animals can sleep while opening one eye and closing one eye.Human have a similar state unihemispheric slow-wave sleep.Sleeping with one eye open and one eye closed is the
32、 result of animal evolution. Too-deep sleep may make them pay a very heavy price. As a result, some animals have evolved the ability of being half awake and half asleep.Most people think : sleeping and waking are two independent states. The first-night effect: an alert to strange environmentThe unin
33、terrupted consciousness, human cant sleep with one eye open and one eye closed, animal could, while human have unihemispheric slow-wave sleep.Fr om the intermediate va lue t heor em of the continu ous fu nction, ther e ex ists 0 0, 2 ， su ch that 0( ) 0h =, that i s , 00( ) ( )fg =， because 00( ) ( ) 0fg = ther efo r e 00( ) ( ) 0fg =. The four feet of the table can touch the floor at the same time.Start Dinner
