Why weren't continuous functions defined as Darboux functions? – math.stackexchange.com #JHedzWorlD

When we were in primary school, teachers showed us graphs of ‘continuous’ functions and said something like

“Continuous functions are those you can draw without lifting your pen”

With this in my mind I remember thinking (something along the lines of)

“Oh, that must mean that if the function takes $2$ values $f(y)<f(z)$ then for every $c$ between $f(y), f(z)$ there must be some $x, (y<x<z)$ such that $f(x)=c$”

And that’s what I thought a continuous function was. But then the $epsilon$-$delta$ definition appeared, which put a more restrictive condition on what a continuous function was.

So, my question is, given the fact that Darboux functions “seem continuous” (in some subjective sense, I guess), why wasn’t this used as the definition of continuity? More generally, how did today’s (analytical) definition of continuity appeared?

You can’t actually draw many Darboux functions without lifting your pen from paper. The extreme example is the Conway base 13 function, which takes every value in every interval and hence is a Darboux function. You can’t however begin to even imagine what this function looks like, let alone draw it with a pen.

So you may want to require the function to be bounded. However this still fails, since you can get functions like,

$$ f(x) = begincases sin (1/x), & textif x neq 0 0 & textif x = 0 endcases $$

Which again, you can’t draw. So one idea may be to require functions $f:[a,b] rightarrow Bbb R$ to be of bounded variation, meaning that,

$$ supleftf(x_i)-f(x_i-1) < infty $$

Intuitively, this ‘measures’ the graph of the line $(x,f(x)) mid x in [a,b] $ and requires it to be finite. Then we intuitively should get a finite length curve which we can actually draw without lifting your pen.

It turns out however, that Darboux functions which have bounded variations are actually continuous. So in attempt to define continuity in a more intuitive way, we have found a more restrictive definition.

Even worse, this still isn’t enough. One can show that the function,

$$ f(x) = begincases x^3sin (1/x), & textif x neq 0 0 & textif x = 0 endcases $$

is continuous and has bounded variation, but you can’t really draw it. Note this function is also differentiable and has continuous first derivative. I’ll spare you the details, but using similar ideas we can also construct infinitely differentiable functions of bounded variation, which you can’t draw on paper.

From this, I think you can see why we don’t try to model continuity off the intuitive definition. Instead, we adopt the usual definition because it’s a much more useful and interesting class of functions to work with.

The Darboux definition does not correspond very well with our intuition about continuity. For example, the Conway function takes on every value in every interval, and is therefore Darboux. However it is not continuous, and I don’t think we want it to be continuous, because it certainly doesn’t agree with your teacher’s definition of drawing without lifting your pencil.

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Why weren't continuous functions defined as Darboux functions? – math.stackexchange.com #JHedzWorlD