0001000101000feActProject.ACT000102000eContinuity.EAC0100000070e3ÉÐ<Ì$¨[ Continuity\AuthoráàœBarry Kissane School of Education Murdoch University Murdoch, WA, Australia kissane@murdoch.edu.au http//wwwstaff.murdoch. edu.au/~kissane ÿ[[ 1. Objective:[To explore graphically [and numerically some [discontinuous functions to[understand how they are[represented by ICT.[[2. Exploration 1[Continuous functions have[smooth graphs with no[holes or jumps.[[On computers and on [the ClassPad 300, you[need to be careful with[graphs. For example, [consider the graph of [y=x2-1x-1.[Tap the graph window[below to[\Graph the functionÌʬGraph2DT †Graph3Dh †LISTSYSt4N†Modify ¨$ †STATCALC ÌN†STATSYS Ô\N†Sequence0, †Sheet\| †Sheet3DØ| †SolveEqT†SolveLwrX †SolveUprd †StupFLG1p(†StupList˜D †StupPictÜ$ †ViewWind †y1 H†ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä´ †äÀ †äÌ †äØ †ää †äð †ä ü †ä! †ä" †ä# †ä$, †ä%8 †ä&D †ä'P †ä(\ †ä)h †ä*t †ä+€ †ä,Œ †ä-˜ †ä.¤ †ä/°¤ †ä0T †ä1` †ä2l †ä3x †ä4„ †ä5 †äEœ †äF¨ †äH´ †äIÀ †äJÌ †äKØ †äLä †äMð †äNü †äO †äP †äQ †äR, †äS8 †äTD †ä]P †ä^T †ä_X †ä`\ †äa` †äbd †ä”h †ä•t †äÍ€ †äÎŒ †äИ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â, aseq_histbNewFolde systemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,  0@PSheet1úju^Sheet2úju^Sheet3úju^Sheet4úju^Sheet5úju^SheetSheet3D 0@PSheet1èju^Sheet2èju^Sheet3èju^Sheet4èju^Sheet5èju^SheetSheet3D i™ ™Ð<Ì$ $  3x(x^2-1)/(x-1)`P`P`p`p ™€`€`DîDîDî ™ $0<HT`lx„œ¨´ÀÌØäðü ,8DP\ht€Œ˜¤°¼ÈÔàì ™ ™  ™0 ™@ ™P ™` ™p ™€ ™€  00@@PP``pp€€` ™Q[[The graph appears to be[a line. However, trace[the graph (on the [Analysis menu) to see a[problem.[[Q1. What is the problem?[[Q2. Why did the graph[not show the problem?[[To understand the [problem better, examine[the table of values for[the function (tap the [graph and then the top[left icon on the toolbar)[[When finished, Close the[table from the î  menu[ [3. Exploration 2[To avoid the previous[problem, turn off the[grid: In the î  menu,[choose Settings, Setup [and Graph Format to [turn off the Grid Points.[Then tap below to graph[ the function.[\Graph the functionÌÊlGraph2DT †Graph3Dh †LISTSYSt4N†Modify ¨$ †STATCALC ÌN†STATSYS Ô\N†Sequence0, †Sheet\| †Sheet3DØ| †SolveEqT†SolveLwrX †SolveUprd †StupFLG1p(†StupList˜D †StupPictÜ$ †ViewWind †y1 H†ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä´ †äÀ †äÌ †äØ †ää †äð †ä ü †ä! †ä" †ä# †ä$, †ä%8 †ä&D †ä'P †ä(\ †ä)h †ä*t †ä+€ †ä,Œ †ä-˜ †ä.¤ †ä/°d †ä0 †ä1 †ä2, †ä38 †ä4D †ä5P †äE\ †äFh †äHt †äI€ †äJŒ †äK˜ †äL¤ †äM° †äN¼ †äOÈ †äPÔ †äQà †äRì †äSø †äT †ä] †ä^ †ä_ †ä` †äa  †äb$ †ä”( †ä•4 †äÍ@ †äÎL †äÐX †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â, aseq_histbNewFolde systemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,  0@PSheet1úju^Sheet2úju^Sheet3úju^Sheet4úju^Sheet5úju^SheetSheet3D 0@PSheet1èju^Sheet2èju^Sheet3èju^Sheet4èju^Sheet5èju^SheetSheet3D i™ ™Ð<Ì$ $  3x(x^2-1)/(x-1)`P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜`  $0<HT`lx„œ¨´ÀÌØäðü`````````€` ™Q[[This time the graph[shows the point of [discontinuity at x=1 by a[hole.[[Zoom In several times to[see that the hole is[always visible (you may[need to pan the screen[to keep the hole visible)[[Check that the table [still shows the point of[discontinuity.[[Now Zoom Quick Standard[to change the screen[ dimensions.[[Q3. Explain why the hole[is missing again.[[4. Exploration 3[Another way in which a[function can be [discontinuous is a jump[discontinuity, where the [values jump suddenly. An[example is the greatest[integer function, which[gives the greatest[integer less than a[particular value:[R intg(2.56)R2R[Change the value above[and press EXE to see [how the function works.[Try negative values too.[[Examine the graph of this[function by tapping the[graph window below:[\ Graphing IntgÌÊäGraph2DT †Graph3Dh †LISTSYSt4N†Modify ¨$ †STATCALC ÌN†STATSYS Ô\N†Sequence0, †Sheet\| †Sheet3DØ| †SolveEqT†SolveLwrX †SolveUprd †StupFLG1p(†StupList˜D †StupPictÜ$ †ViewWind †y1H†ä4 †ä@ †äL †äX †äd †äp †ä| †äˆ †ä” †ä  †ä¬ †ä¸ †äÄ †äÐ †äÜ †äè †ä ô †ä! †ä" †ä# †ä$$ †ä%0 †ä&< †ä'H †ä(T †ä)` †ä*l †ä+x †ä,„ †ä- †ä.œ †ä/¨ä †ä0Œ †ä1˜ †ä2¤ †ä3° †ä4¼ †ä5È †äEÔ †äFà †äHì †äIø †äJ †äK †äL †äM( †äN4 †äO@ †äPL †äQX †äRd †äSp †äT| †ä]ˆ †ä^Œ †ä_ †ä`” †äa˜ †äbœ †ä”  †ä•¬ †ä͸ †äÎÄ †äÐÐ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â, aseq_histbNewFolde systemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,  0@PSheet1úju^Sheet2úju^Sheet3úju^Sheet4úju^Sheet5úju^SheetSheet3D 0@PSheet1èju^Sheet2èju^Sheet3èju^Sheet4èju^Sheet5èju^SheetSheet3D i™ ™Ð<Ì$ $  3xà–(x)`P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜PìPìPì ™ $0<HT`lx„œ¨´ÀÌØäðü ,8DP\ht€Œ˜¤°¼ÈÔàìø(4@LXdp|ˆ” ¬¸ÄÐÜ ™ ™ ™ ™ ™ ™ ™ ™ ™ 0@P`p€ 0@P`p€` ™Q[Trace the graph to see[the jump discontinuities.[[Q4. At which points is[Intg(x) not continuous?[[Q5. Explain what is[wrong with the graph.[[You may find it useful to[make a table of values[as well as a graph to[see the jumps.[[5. Exploration 4 [Computers draw graphs[by plotting points and [connecting them with[lines. To avoid showing[jump discontinuities[incorrectly, change the[way graphs are drawn: [In the î  menu,[choose Settings, Setup [and Graph Format; then[in the Special menu, [change Draw Type from[Connect to Plot. Now[tap below to graph the[function again.[\ Step functionÌÊìGraph2D@ †Graph3DT †LISTSYS`4N†Modify ”$ †STATCALC ¸N†STATSYS À\N†Sequence, †SheetH| †Sheet3DÄ| †SolveEq@†SolveLwrD †SolveUprP †StupFLG1\(†StupList„D †StupPictÈ$ †ViewWindì †y1H†ä †ä, †ä8 †äD †äP †ä\ †äh †ät †ä€ †äŒ †ä˜ †ä¤ †ä° †ä¼ †äÈ †äÔ †ä à †ä!ì †ä"ø †ä# †ä$ †ä% †ä&( †ä'4 †ä(@ †ä)L †ä*X †ä+d †ä,p †ä-| †ä.ˆ †ä0” †ä1  †ä2¬ †ä3¸ †ä4Ä †ä5Ð †äEÜ †äFè †äHô †äI †äJ †äK †äL$ †äM0 †äN< †äOH †äPT †äQ` †äRl †äSx †äT„ †ä] †ä^” †ä_˜ †ä`œ †äa  †äb¤ †ä”¨ †ä•´ †äÍÀ †äÎÌ †äÐØ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â,Â, aseq_histbNewFolde systemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,Âsystemä]listsystemä^]=system,  0@PSheet1úju^Sheet2úju^Sheet3úju^Sheet4úju^Sheet5úju^SheetSheet3D 0@PSheet1èju^Sheet2èju^Sheet3èju^Sheet4èju^Sheet5èju^SheetSheet3D i™ ™Ð<Ì$ $  3xà–(x)`P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜``` ™P[[Trace the function and [zoom in to see the jump[discontinuities are still[there, even though the[graph is now correct.[[Q6. Why does the graph[no longer show the[discontinuity incorrectly?[[ 6. Exercises[Examine any discontinuity[of these functions:[[(i) y = Int(x)[ (ii) y = 1/x[(iii) y=x2-4x+2[ \Answersáà´¨Q1. There is an error for x=1, as the function is not defined at that point. So the function is discontinuous at x=1. Q2. There is a grid point plotted at (1,1), hiding the hole in the graph. Q3. The choice of Standard does not allow the precise point x=1 to be plotted. Q4. At integral values of x, such as x = -2, -1, 0 , 1, 2, ... Q5. The graph should not be joined (almost) vertically at integral values of x. Q6. The Plot choice for Draw Type means that individual points are plotted, but they are nolonger joined. Exercises (i) Has jump discontinuityat integer values of x, except for x=-1. (ii) Has jump discontinuityat x =0 (iii) Has discontinuity at x=-2. ÿ[[Don't forget to restore[the Graph Format[settings to their original[values when you've [ finished.[[Return to Menu without[saving[eActangles Ô  $0<HT`lx„Œœœœœœœœœ œ@œ`œu8Åarea D $0<HT`lx„œ¨´ÀÌØäY5Ya`HyTe`a€I’™(X6f“‘G‰H 3…U2F`7sC)E(8xGEaRB'HgC63@TG…QˆC'S8sP06C† 62‘„)GU2AX7 T X6b”3Sm_angle m_area volume D $0<HT`lx„œ¨´ÀÌØäY5Ya`HyTe`a€I’™(X6f“‘G‰H 3…U2F`7sC)E(8xGEaRB'HgC63@TG…QˆC'S8sP06C† 62‘„)GU2AX7 T X6b”3Sx D $0<HT`lx„œ¨´ÀÌØäœ —œ ˜œ ™¬ý     ( 2 < P d x Œ   ¯xœyœyœy