[contravariant]

He didn't seem to have realized fully what a 'graph' was. He probably went with the 'standard procedure' for drawing a graph (pick a x, and draw a point f(x) places above the x-axis, repeat).

He later realized that functions are a type of relation, (which is made explicit with the notation "y = f(x)") and that the graph is the set of points that satisfy that relation. Of course if you realize that you can also generalize to g(x) = f(y), f(x,y) = 0 etc. Which he seems to have done.

It also seems he hadn't completely realized that a graph is merely a collection of points, not a nice line-like object. This last is more explicit in non-continuous functions, but you don't usually see those in high school.

[xpinguin]

> It also seems he hadn't completely realized that a graph is merely a collection of points, not a nice line-like object. This last is more explicit in non-continuous functions, but you don't usually see those in high school.

Well, I find whole thing rather controversial, so to say. Depends on what do one mean by "collection" and by "point", eg. if we take point literally: (x, y) \in RxR, then we have to assume that "collection" is uncountable (which seems counterintuitive, esp. from the school student's pov). OTOH, one could treat "point" as representation of that point, eg. pencil mark, so that point is no longer single element, but rather an uncountable subset containing the point of interest, along with its neighbourhood => our "collection" becomes countable, so we could think about it as bunch of indexed points (much more intuitive, isn't it?), shifting our attention towards mapping's representation from the mapping itself.

tl;dr Treating graph as a collection of points is no more correct/better, compared to treating it as a continous line. Different problems require different approaches and different "angles" of view. Barely an insight from the author - the most of the mathematics is about abstractions and pattern matching.

[rsy96]

> Well, I find whole thing rather controversial, so to say. Depends on what do one mean by "collection" and by "point", eg. if we take point literally: (x, y) \in RxR, then we have to assume that "collection" is uncountable (which seems counterintuitive, esp. from the school student's pov). OTOH, one could treat "point" as representation of that point, eg. pencil mark, so that point is no longer single element, but rather an uncountable subset containing the point of interest, along with its neighbourhood => our "collection" becomes countable, so we could think about it as bunch of indexed points (much more intuitive, isn't it?), shifting our attention towards mapping's representation from the mapping itself.

The whole paragraph does not make sense to me. For example, how is a collection uncountable somehow unintuitive? Why should a point not be a single element, but a subset or a pencil mark? Are you confusing the theoretical graph with the graph actually drawn (which has finite width)?

[xpinguin]

There are two points in that paragraph:

- enumerable (countable) sets are more intuitive to reason about

- no matter, how do you comprehend the function, you'll end up drawing it by means of composing continuous chunks

> Are you confusing the theoretical graph with the graph actually drawn (which has finite width)? Probably. From my point of view, graph is the something that is drawn/plotted, whilst relation/mapping being something that you've called "theoretical graph". So I'd question, how treating graph as a bunch of points is superior concept, giving that one could easily slip into substituting actual mapping's points by their graphical representation, for the sake of simplicity, thus possibly hiding mapping's behaviour from own mind.

PS Nevertheless, I do now understand, that the whole thing author has meant to say, was: "Given x = f(y), it is not only some explicit line on a plane, but also a mapping f: (x, y) \in RxR (which also could be drawn, btw)".

The whole subthread is more of a dialectical excercise, with the definition of "graph" not being synchronized among participants :)

[kaitai]

Treating a graph as a collection of points is certainly more correct and better if the condition you are trying to satisfy gives a non-continuous set of points. If you can only deal with continuous functions you will be lost in these more interesting cases.