point in math and continuous function: Is there more than smaller to the point?

Ryan$

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Jan 25, 2019
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Hi guys, I have to confusions which please illustrate for me them in a simple manner.
first is there more than smaller to the point? I mean if I imagine a point .. then I can also imagine a point inside the point that I imagined which it's smaller than what I imagined .. so?

second question, is continuous function means that at everypoint there's a possibility that the values of function over it will be changed? thanks alot

and who said that at every point I have value? maybe at semi-point I have value? who corresponded that a value corresponded to a point and not to semi-point?! thanks alot
 
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Dr.Peterson

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No, if you correctly imagine a point, then you can't imagine anything smaller! A point is to be thought of as a mere location, with no size at all. Yes, it's hard to imagine that; but if you imagine a size, then you are imagining wrongly, just as you would be wrong to imagine a vacuum from which you could suck more air than you already did to form it.

As for functions, a function isn't something that changes; it is a particular relationship between an input and an output. For any give input, the output is fixed. So it's not at all clear what you are thinking. A continuous function (leaving out some technical details) is one whose graph can be drawn in one continuous line, without ever lifting your pencil. It has to do with how values of x close to one another affect values of y -- if x doesn't change much, y shouldn't change much. (You need calculus to really express this correctly.)
 

Ryan$

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Jan 25, 2019
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133
Wow convinced me about point in away that I cant say anything :) 😊
Another thing then why we assume that value is corresponded to a point and not for example a value corresponded to semi-point? I mean maybe value occupy two points at one time? Who claims that value number is corresponded to one point at a time?
 

Dr.Peterson

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Math is invented so that it represents the concepts we are trying to represent, as simply as possible. Assuming that each number corresponds to one location on a number line agrees with what we expect from generalizations of the real world, and makes calculations easy, so we go with that (and have for a very long time).

Now, perhaps you could invent a new kind of geometry in which there was such a thing as a "semi-point" and there were two numbers in the same place on a number line; but you probably couldn't get anyone to try using it, because it would be too cumbersome with no benefits. That's not to say it would necessarily be "wrong".

Mathematicians do invent new mathematical objects or systems, just to see what will happen! And sometimes the results turn out to be useful, even if they weren't trying to make something that corresponds to the real world. But if what they invent is not either useful or interesting, their paper will just gather dust, even if it's perfectly valid, because there will be no motivation for anyone to pursue it further.
 
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