连续性与开集（工作进展）

几何画板及绘图内容加载中，请稍等……

2018-06-19 13:07

函数微积分几何作图轨迹

所属专题 Mapping Diagrams to Visualize Complex Analysis

本范例演示了连续函数与开集在数学中的关系，通过动态图形展示函数连续性的直观理解。涉及的核心数学概念包括连续函数、开集、ε-δ定义等。作图要点运用了GeoGebra的Sequence, If, Circle, Line, Vector, Point, Locus, Textfield, PolyLine, Zip等命令与对象，通过滑块动画和轨迹展示数学概念。适合高中及以上数学学习者和教师使用。

GeoGebra 命令及参数

Sequence(0, m) → M
If(e ≟ 1, z_0 - fz_0, 0) → align
Circle(z_0, δ) → δcirc
Line(O, ZY1) → Yax
Line(O, ZX1) → Xax
Vector(z_0, Fz1_#) → vecf_#
Point(δcirc) → A0
Locus(PtfA0, A0) → fδcirc
If(e ≟ 1, (x(z_0), y(z_0), ht), (x(fz_0), y(fz_0), ht)) → Alignfz_0
Vector(z_0, Alignfz_0) → u
Textfield(F) → InputBox1
Circle(L, ε, Vector[(0, 0, 1)]) → εcir
PolyLine((3.016516497546083, -0.19504792639786206), (NaN, NaN), (2.8444653716061468, 0.23507988845198102), (2.9304909345761145, -1.9908315533959564), (3.070282474402315, -1.9155591857972332), (3.16706123274353, -1.8725464043122488), (3.306852772569729, -1.7972740367135265), (3.392878335539697, -1.7542612552285421), (3.5541762661083895, -1.668235692258573), (3.6294486337071117, -1.6037165200310965), (3.6617082198208504, -1.5607037385461122), (3.6939678059345873, -1.474678175576144), (3.726227392048327, -1.3671462218636832), (2.8229589808636546, 0.2780926699369656), (3.7692401735333108, -1.2596142681512232), (3.7799933689045564, -1.1950950959237459), (3.790746564275802, -1.1305759236962694), (3.790746564275802, -1.0660567514687929), (3.790746564275802, -1.023043969983809), (3.790746564275802, -0.9585247977563325), (3.790746564275802, -0.915512016271349), (3.7584869781620633, -0.8509928440438725), (3.726227392048327, -0.8402396486726265), (3.672461415192096, -0.7972268671876424), (2.736933417893686, 0.3856246236494255), (3.618695438335866, -0.7864736718163963), (3.5864358522221274, -0.7649672810739041), (3.5326698753658983, -0.7649672810739041), (3.392878335539697, -0.7434608903314117), (3.3283591633122196, -0.7434608903314117), (3.285346381827236, -0.7434608903314117), (3.242333600342252, -0.7434608903314117), (3.1778144281147758, -0.7434608903314117), (3.091788865144806, -0.7434608903314117), (3.0380228882885763, -0.7434608903314117), (2.6724142456662094, 0.4716501866193942), (2.995010106803592, -0.7327076949601656), (2.9627505206898532, -0.7004481088464274), (2.941244129947362, -0.6681885227326891), (2.9304909345761145, -0.6359289366189509), (2.919737739204869, -0.5499033736489827), (2.8982313484623776, -0.49613739679275215), (2.8982313484623776, -0.45312461530776793), (2.8982313484623776, -0.4101118338227836), (2.8982313484623776, -0.3563458569665532), (2.9089845438336233, -0.3025798801103228), (2.5756354873249943, 0.5146629681043781), (2.9304909345761145, -0.24881390325409247), (2.973503716061099, -0.2273075125116003), (3.0057633021748376, -0.21655431714035422), (3.0380228882885763, -0.19504792639786206), true, (2.3390651891575795, 0.6329481171880855), (2.00571613264895, 0.772739657014285), (1.9089373743077351, 0.8157524384992694), (1.8659245928227508, 0.8157524384992694), (1.7906522252240278, 0.8157524384992694), (3.016516497546083, -0.19504792639786206), (1.6078479039128435, 0.8157524384992694), (1.468056364086644, 0.8157524384992694), (1.4250435826016596, 0.8157524384992694), (1.3712776057454297, 0.7834928523855311), (1.3067584335179527, 0.7512332662717933), (1.263745652032968, 0.7189736801580546), (1.1884732844342458, 0.6759608986730699), (1.1239541122067689, 0.6329481171880855), (1.0701881353505387, 0.6114417264455932), (1.0271753538655544, 0.5791821403318546), (2.9842569114323463, -0.1090223634278934), (0.9519029862668318, 0.5254161634756246), (0.9088902047818466, 0.49315657736188634), (0.8443710325543705, 0.4393906005056559), (0.8013582510693862, 0.3963778190206716), (0.7475922742131553, 0.3318586467931953), (0.44650280381826457, 0.041522371769551536), (0.3389708501058033, 0.009262785655813288), (0.24219209176458856, -0.02299680045792485), (0.1239069426808812, -0.055256386571663096), (0.08089416119589687, -0.06600958194290918), (2.973503716061099, -0.03374999582917093), (0.027128184339666017, -0.09826916805664732), (-0.0373909878878107, -0.13052875417038556), (-0.06965057400154828, -0.1735415356553699), (-0.12341655085777914, -0.21655431714035422), (-0.17718252771401022, -0.25956709862533844), (-0.2309485045702402, -0.31333307548156886), (-0.27396128605522496, -0.36709905233779927), (-0.3169740675402095, -0.4208650291940297), (-0.35998684902519384, -0.4746310060502601), (-0.39224643513893254, -0.5283969829064905), (2.9519973253186076, 0.030769176398305453), (-0.435259216623916, -0.6036693505052131), (-0.45676560736640814, -0.6574353273614431), (-0.4782719981089003, -0.7434608903314117), (-0.4782719981089003, -0.7864736718163963), (-0.4782719981089003, -0.8294864533013804), (-0.4782719981089003, -0.8940056255288564), (-0.4460124119951616, -0.9370184070138412), (-0.35998684902519384, -0.9907843838700712), (-0.2417016999414865, -1.033797165355055), (-0.12341655085777914, -1.087563142211285), (2.941244129947362, 0.0630287625120437), (-0.005131401774072231, -1.1090695329537772), (0.07014096582465079, -1.1090695329537772), (0.16691972416586554, -1.1090695329537772), (0.2206857010220964, -1.1090695329537772), (0.30671126399206505, -1.1090695329537772), (0.3497240454770498, -1.1090695329537772), (0.40349002233328024, -1.1090695329537772), (0.4787623899320028, -1.1090695329537772), (0.5862943436444641, -1.1413291190675154), (0.6293071251294482, -1.1520823144387615), (2.9089845438336233, 0.10604154399702803), (0.6615667112431867, -1.1628355098100076), (0.6938262973569245, -1.1843419005524998), (0.7260858834706632, -1.2058482912949924), (0.7475922742131553, -1.2488610727799767), (0.7690986649556475, -1.2918738542649615), (0.7906050556981397, -1.3563930264924375), (0.8121114464406318, -1.3886526126061753), (0.8228646418118784, -1.4424185894624058), (0.833617837183124, -1.474678175576144), (0.833617837183124, -1.5176909570611283), (2.8982313484623776, 0.15980752085325844), (0.8443710325543705, -1.5822101292886044), (0.8551242279256162, -1.6252229107735887), (0.8658774232968627, -1.668235692258573), (0.8766306186681083, -1.7004952783723113), (0.8873838140393548, -1.7757676459710343), (0.9519029862668318, -1.8510400135697567), (1.0809413307217848, -1.9263123811684797), (1.1884732844342458, -1.969325162653464), (1.2960052381467062, -1.9800783580247099), (1.3927839964879218, -1.9908315533959564), (2.86597176234864, 0.20282030233824277), (1.4250435826016596, -2.0015847487672023), (1.4895627548291361, -2.012337944138449), (1.5970947085415979, -2.0230911395096953), (1.6831202715115665, -2.0230911395096953), (1.8014054205952743, -2.0230911395096953), (1.9304437650502273, -2.0230911395096953), (2.0379757187626883, -2.0445975302521875), (2.2207800400738718, -2.0445975302521875), (2.4358439474987947, -2.0445975302521875), (2.607895073438732, -2.0445975302521875)) → Domain
Point(Domain) → B
Locus(FBpt, B) → loc1
Zip(((2 * π) * k / m), k, M) → t_*
Vector((0, 0, 0), (0, 0, ht)) → totarg
PolyLine((2.3393495483444755, -0.19967392328402625), (NaN, NaN), (2.0823086774376773, 0.2144474798435947), (2.38218969349561, -0.45671479419082495), (2.3679096451118986, -0.41387464903969184), (2.353629596728186, -0.3710345038885583), (2.353629596728186, -0.31391431035371464), (2.353629596728186, -0.2425140684351589), (2.353629596728186, -0.18539387490031503), (2.353629596728186, -0.12827368136547038), true, (2.053748580670256, 0.25728762499472735), (1.982348338751701, 0.30012777014586045), (1.939508193600568, 0.314407818529572), (1.8823880000657236, 0.3286878669132831), (1.8252678065308783, 0.3286878669132831), (1.7253074678449027, 0.3572479636807049), (1.611067080775216, 0.3572479636807049), (1.5539468872403726, 0.3572479636807049), (1.482546645321817, 0.3572479636807049), (2.3393495483444755, -0.19967392328402625), (1.425426451786973, 0.3572479636807049), (1.368306258252129, 0.3572479636807049), (1.296906016333574, 0.3429679152969942), (1.2540658711824402, 0.3286878669132831), (1.1969456776475966, 0.28584772176214934), (1.1398254841127522, 0.25728762499472735), (1.0827052905779082, 0.2001674314598836), (1.0398651454267749, 0.18588738307617206), (0.9970250002756416, 0.1716073346924612), (0.96846490350822, 0.12876718954132876), (2.3107894515770533, -0.1568337781328926), (0.9256247583570867, 0.08592704439019566), (0.9113447099733756, 0.04308689923906234), (0.8827846132059534, 0.0002467540879289021), (0.8399444680548203, -0.042593391063203964), (0.8256644196711092, -0.0854335362143375), (0.7971043229036874, -0.17111382651660392), (0.7971043229036874, -0.29963426197000353), (0.7971043229036874, -0.41387464903969184), (0.811384371287398, -0.47099484257453605), (0.811384371287398, -0.5281150361093796), (2.2965094031933426, -0.11399363298175971), (0.811384371287398, -0.5852352296442245), (0.811384371287398, -0.6423554231790696), (0.8256644196711092, -0.6851955683302025), (0.8399444680548203, -0.7280357134813358), (0.8542245164385314, -0.7708758586324693), (0.8685045648222427, -0.8279960521673131), (0.8685045648222427, -0.8851162457021575), (0.8827846132059534, -0.9279563908532913), (0.8970646615896645, -0.9707965360044242), (0.8970646615896645, -1.0279167295392668), (2.2679493064259213, -0.07115348783062618), (0.9113447099733756, -1.0707568746903997), (0.9256247583570867, -1.1135970198415321), (0.9541848551245085, -1.1707172133763775), (0.96846490350822, -1.2135573585275106), (0.9970250002756416, -1.256397503678643), (1.0255850970430642, -1.3420777939809079), (1.0827052905779082, -1.4134780358994627), (1.1398254841127522, -1.442038132666885), (1.182665629263886, -1.4848782778180196), (1.225505774415018, -1.5134383745854418), (2.239389209658499, -0.014033294295781873), (1.2540658711824402, -1.5705585681202872), (1.2540658711824402, -1.6419588100388423), (1.3111860647172855, -1.6847989551899751), (1.368306258252129, -1.6847989551899751), (1.425426451786973, -1.6990790035736856), (1.482546645321817, -1.6990790035736856), (1.525386790472949, -1.7133590519573971), (1.5825069840077939, -1.7276391003411078), (1.6396271775426383, -1.7276391003411078), (1.7253074678449027, -1.7419191487248176), (2.2108291128910778, 0.02880685085535145), (1.7824276613797463, -1.7419191487248176), (1.8538279032983014, -1.7419191487248176), (1.9109480968331458, -1.7419191487248176), (1.9680682903679902, -1.7419191487248176), (2.025188483902833, -1.7419191487248176), (2.0823086774376773, -1.7419191487248176), (2.1537089193562333, -1.7419191487248176), (2.2251091612747884, -1.7419191487248176), (2.2965094031933426, -1.7419191487248176), (2.3679096451118986, -1.7276391003411078), (2.1394288709725218, 0.10020709277390677), (2.4107497902630315, -1.6990790035736856), (2.4535899354141635, -1.6419588100388423), (2.4964300805652964, -1.61339871327142), (2.5249901773327186, -1.5705585681202872), (2.55355027410014, -1.5277184229691543), (2.5678303224838506, -1.4848782778180196), (2.596390419251272, -1.442038132666885), (2.6820707095535385, -1.1135970198415321), (2.6820707095535385, -1.0421967779229784), (2.6820707095535385, -0.9850765843881348), (2.1108687742051004, 0.1573272863087501), (2.6535106127861163, -0.91367634246958), (2.6392305644024048, -0.8422761005510242), (2.596390419251272, -0.7423157618650471), (2.582110370867562, -0.6994756167139142), (2.5678303224838506, -0.6566354715627802), (2.5249901773327186, -0.628075374795358), (2.4821500321815857, -0.6137953264116474), (2.439309887030453, -0.599515278027936), (2.39646974187932, -0.5709551812605136), (2.38218969349561, -0.5281150361093796)) → finvU
Point(finvU) → E
Locus(E', E) → TarU
Zip((δ * ℯ^((t * ί))) + z_0, t, t_*) → z_*
Zip(f(z) + align, z, z_*) → fz_*
Zip((x(z), y(z), ht), z, fz_*) → fpt_*
Zip(Vector[A, B], A, z_*, B, fpt_*) → fvect_*
Vector(z_0, A0) → v
Vector(L, L + (0, ε, 0)) → εvec

AI 绘图提示语参考

绘制一个函数的连续性和开集关系图。首先画出x轴和y轴，然后绘制一个点z₀和一个以z₀为中心、半径为δ的圆。接着画出函数f在z₀处的图像，并用箭头表示函数值的变化。最后，画出一个以ε为半径的圆，表示ε-δ定义中的ε值。整个图形展示了连续函数和开集的概念。

以上提示语为根据原始绘图命令整理而来，仅供参考。