From: ram@zedat.fu-berlin.de   
      
   William Hyde  wrote or quoted:   
   >Stefan Ram wrote:   
   >>William Hyde  wrote or quoted:   
   >>>"A continuous function is one whose inverse maps open sets to open sets"   
   >>It might be better to ask that the preimages of open sets come   
   >>out open, because if you think about the sine function, you   
   >>can still use that test on it even though it does not have an   
   >>inverse, since that wouldn't be single-valued.   
   >We're getting well beyond the limits of my memory here.   
      
     Oh, that's not really a big deal. The whole point is just that   
     every function in math, whether it's topology or whatever else,   
     has to be set up so each x only matches with one y.   
      
     Figure 1 shows that with the sine function.   
      
        |      .+++.   
        |     .+   ^.   
      y |<---.+     ^.   
        |    +|      +   
        |   | |       +   
        |  .' |        .   
        |  |  |        +   
        | .'  |         -   
        | +   |         |   
        |.    |          -   
        |+    |          |   
        |'    |           -   
   +++++|++++++++++++++++++   
              x   
      
     But the other way around, a lot of functions end up giving you a   
     bunch of possible values, so you no longer have a proper "inverse   
     function".   
      
     Figure 2 shows that again with the sine function.   
      
        |      .+++.   
        |     .+   ^.   
      y |----.+-----^.   
        |    +|     |+   
        |   | |     | +   
        |  .' |     |  .   
        |  |  |     |  +   
        | .'  |     |   -   
        | +   |     |   |   
        |.    |     |    -   
        |+    |     |    |   
        |'    V     V     -   
   +++++|++++++++++++++++++   
             x_0   x_1   
      
     You can still talk about points like "x_0" and "x_1" in the   
     figure, but they're just "preimages", not actual outputs of   
     an "inverse function".   
      
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