Home » Archimedes archive » Acorn User » AU 1994-10.adf » !StarInfo_StarInfo » Wareham/Hard
Wareham/Hard
This website contains an archive of files for the Acorn Electron, BBC Micro, Acorn Archimedes, Commodore 16 and Commodore 64 computers, which Dominic Ford has rescued from his private collection of floppy disks and cassettes.
Some of these files were originally commercial releases in the 1980s and 1990s, but they are now widely available online. I assume that copyright over them is no longer being asserted. If you own the copyright and would like files to be removed, please contact me.
| Tape/disk: | Home » Archimedes archive » Acorn User » AU 1994-10.adf » !StarInfo_StarInfo |
| Filename: | Wareham/Hard |
| Read OK: | ✔ |
| File size: | 0427 bytes |
| Load address: | 0000 |
| Exec address: | 0000 |
File contents
On the other hand once the definition is provided the theorem ceases to be obvious, since the result it asserts is quite different from the defining property and to show that the first follows the second is not a trivial task. If it is objected that we should take as definition some property designed to make the proof simpler, the answer is that there are many other 'obvious' and important properties of coninuous functions and no definition simplifies them all simultaneously. We might of course lump together everything we want of a continuous function, and call a function continuous whenever it has these properties. Apart from the crudity and clumsiness of such a procedure, we should thereby entirely obscure the fact that all such properties in fact flow from one simple basic one; we should lose all insight into the relative depths of the properties and into the nature of their interconnections; we might even unwittingly include properties that were subtly inconsistent; and it would take too long to decide whether a given function was continuous.
00000000 4f 6e 20 74 68 65 20 6f 74 68 65 72 20 68 61 6e |On the other han| 00000010 64 20 6f 6e 63 65 20 74 68 65 20 64 65 66 69 6e |d once the defin| 00000020 69 74 69 6f 6e 20 69 73 20 70 72 6f 76 69 64 65 |ition is provide| 00000030 64 20 74 68 65 20 74 68 65 6f 72 65 6d 20 63 65 |d the theorem ce| 00000040 61 73 65 73 20 74 6f 20 62 65 20 6f 62 76 69 6f |ases to be obvio| 00000050 75 73 2c 20 73 69 6e 63 65 20 74 68 65 20 72 65 |us, since the re| 00000060 73 75 6c 74 20 69 74 20 61 73 73 65 72 74 73 20 |sult it asserts | 00000070 69 73 20 71 75 69 74 65 20 64 69 66 66 65 72 65 |is quite differe| 00000080 6e 74 20 66 72 6f 6d 20 74 68 65 20 64 65 66 69 |nt from the defi| 00000090 6e 69 6e 67 20 70 72 6f 70 65 72 74 79 20 61 6e |ning property an| 000000a0 64 20 74 6f 20 73 68 6f 77 20 74 68 61 74 20 74 |d to show that t| 000000b0 68 65 20 66 69 72 73 74 20 66 6f 6c 6c 6f 77 73 |he first follows| 000000c0 20 74 68 65 20 73 65 63 6f 6e 64 20 69 73 20 6e | the second is n| 000000d0 6f 74 20 61 20 74 72 69 76 69 61 6c 20 74 61 73 |ot a trivial tas| 000000e0 6b 2e 20 49 66 20 69 74 20 69 73 20 6f 62 6a 65 |k. If it is obje| 000000f0 63 74 65 64 20 74 68 61 74 20 77 65 20 73 68 6f |cted that we sho| 00000100 75 6c 64 20 74 61 6b 65 20 61 73 20 64 65 66 69 |uld take as defi| 00000110 6e 69 74 69 6f 6e 20 73 6f 6d 65 20 70 72 6f 70 |nition some prop| 00000120 65 72 74 79 20 64 65 73 69 67 6e 65 64 20 74 6f |erty designed to| 00000130 20 6d 61 6b 65 20 74 68 65 20 70 72 6f 6f 66 20 | make the proof | 00000140 73 69 6d 70 6c 65 72 2c 20 74 68 65 20 61 6e 73 |simpler, the ans| 00000150 77 65 72 20 69 73 20 74 68 61 74 20 74 68 65 72 |wer is that ther| 00000160 65 20 61 72 65 20 6d 61 6e 79 20 6f 74 68 65 72 |e are many other| 00000170 20 27 6f 62 76 69 6f 75 73 27 20 61 6e 64 20 69 | 'obvious' and i| 00000180 6d 70 6f 72 74 61 6e 74 20 70 72 6f 70 65 72 74 |mportant propert| 00000190 69 65 73 20 6f 66 20 63 6f 6e 69 6e 75 6f 75 73 |ies of coninuous| 000001a0 20 66 75 6e 63 74 69 6f 6e 73 20 61 6e 64 20 6e | functions and n| 000001b0 6f 20 64 65 66 69 6e 69 74 69 6f 6e 20 73 69 6d |o definition sim| 000001c0 70 6c 69 66 69 65 73 20 74 68 65 6d 20 61 6c 6c |plifies them all| 000001d0 20 73 69 6d 75 6c 74 61 6e 65 6f 75 73 6c 79 2e | simultaneously.| 000001e0 20 57 65 20 6d 69 67 68 74 20 6f 66 20 63 6f 75 | We might of cou| 000001f0 72 73 65 20 6c 75 6d 70 20 74 6f 67 65 74 68 65 |rse lump togethe| 00000200 72 20 65 76 65 72 79 74 68 69 6e 67 20 77 65 20 |r everything we | 00000210 77 61 6e 74 20 6f 66 20 61 20 63 6f 6e 74 69 6e |want of a contin| 00000220 75 6f 75 73 20 66 75 6e 63 74 69 6f 6e 2c 20 61 |uous function, a| 00000230 6e 64 20 63 61 6c 6c 20 61 20 66 75 6e 63 74 69 |nd call a functi| 00000240 6f 6e 20 63 6f 6e 74 69 6e 75 6f 75 73 20 77 68 |on continuous wh| 00000250 65 6e 65 76 65 72 20 69 74 20 68 61 73 20 74 68 |enever it has th| 00000260 65 73 65 20 70 72 6f 70 65 72 74 69 65 73 2e 20 |ese properties. | 00000270 41 70 61 72 74 20 66 72 6f 6d 20 74 68 65 20 63 |Apart from the c| 00000280 72 75 64 69 74 79 20 61 6e 64 20 63 6c 75 6d 73 |rudity and clums| 00000290 69 6e 65 73 73 20 6f 66 20 73 75 63 68 20 61 20 |iness of such a | 000002a0 70 72 6f 63 65 64 75 72 65 2c 20 77 65 20 73 68 |procedure, we sh| 000002b0 6f 75 6c 64 20 74 68 65 72 65 62 79 20 65 6e 74 |ould thereby ent| 000002c0 69 72 65 6c 79 20 6f 62 73 63 75 72 65 20 74 68 |irely obscure th| 000002d0 65 20 66 61 63 74 20 74 68 61 74 20 61 6c 6c 20 |e fact that all | 000002e0 73 75 63 68 20 70 72 6f 70 65 72 74 69 65 73 20 |such properties | 000002f0 69 6e 20 66 61 63 74 20 66 6c 6f 77 20 66 72 6f |in fact flow fro| 00000300 6d 20 6f 6e 65 20 73 69 6d 70 6c 65 20 62 61 73 |m one simple bas| 00000310 69 63 20 6f 6e 65 3b 20 77 65 20 73 68 6f 75 6c |ic one; we shoul| 00000320 64 20 6c 6f 73 65 20 61 6c 6c 20 69 6e 73 69 67 |d lose all insig| 00000330 68 74 20 69 6e 74 6f 20 74 68 65 20 72 65 6c 61 |ht into the rela| 00000340 74 69 76 65 20 64 65 70 74 68 73 20 6f 66 20 74 |tive depths of t| 00000350 68 65 20 70 72 6f 70 65 72 74 69 65 73 20 61 6e |he properties an| 00000360 64 20 69 6e 74 6f 20 74 68 65 20 6e 61 74 75 72 |d into the natur| 00000370 65 20 6f 66 20 74 68 65 69 72 20 69 6e 74 65 72 |e of their inter| 00000380 63 6f 6e 6e 65 63 74 69 6f 6e 73 3b 20 77 65 20 |connections; we | 00000390 6d 69 67 68 74 20 65 76 65 6e 20 75 6e 77 69 74 |might even unwit| 000003a0 74 69 6e 67 6c 79 20 69 6e 63 6c 75 64 65 20 70 |tingly include p| 000003b0 72 6f 70 65 72 74 69 65 73 20 74 68 61 74 20 77 |roperties that w| 000003c0 65 72 65 20 73 75 62 74 6c 79 20 69 6e 63 6f 6e |ere subtly incon| 000003d0 73 69 73 74 65 6e 74 3b 20 61 6e 64 20 69 74 20 |sistent; and it | 000003e0 77 6f 75 6c 64 20 74 61 6b 65 20 74 6f 6f 20 6c |would take too l| 000003f0 6f 6e 67 20 74 6f 20 64 65 63 69 64 65 20 77 68 |ong to decide wh| 00000400 65 74 68 65 72 20 61 20 67 69 76 65 6e 20 66 75 |ether a given fu| 00000410 6e 63 74 69 6f 6e 20 77 61 73 20 63 6f 6e 74 69 |nction was conti| 00000420 6e 75 6f 75 73 2e 0a |nuous..| 00000427
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