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I was trying to find a definition of the continuity of a function that fits better the unrirgorous definition that "it can be drawed with a pencil in a single stroke" and I came up with this: a function $f : I → R$ is "continuous" iff for any $a, b \in I$ and any $\alpha \in (f(a), f(b))$ there is an $x \in (a, b)$ so that $f(x) = \alpha$. Basically this means that between two points $a$ and $b$ $f$ must assume all values between $f(a)$ and $f(b)$.

This definition is obviously no equivalent to the normal definitions since functions such as $\frac{1}{x}$ and $\tan x$ would not be continuous. But on the other hand they cannot be drawn in a single stroke (kind of). I myself find it a lot more intuitive than the normal definitions.

What I would want to know is:

  • how much sense does this definition make?
  • Is it correct for functions whose domain is connected (so not $\frac1x$, $\tan x$)?

English is not my first language and I don't learn math in English, so I may have used some terms wrongly. Please ask if something doesn't make sense.

Carlos Esparza
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    it is called the intermediate value property. the class of functions satisfying that property is larger than the class of continuous functions. – user251257 Sep 07 '15 at 10:57
  • Note also that the class of functions that comes closest to "can be drawed with a pencil in a single stroke" is the set of Lipschitz-continuous functions. Functions with less continuity can look very uncontinuous. – Lutz Lehmann Sep 07 '15 at 17:18

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Your proposed definition is of functions usually called Darboux functions. For example, every derivative (of a differentiable function) has this property, but there are others as well. For example $$ f(x) = \begin{cases} \sin \frac1x, & x \neq 0 \\ 0, & x = 0. \end{cases} $$

mrf
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  • The Wikipedia article on Darboux functions states: "A Darboux function is a real-valued function f [...] for any two values a and b in the domain of f, and any y between f(a) and f(b), there is some c between a and b with f(c) = y". This doesn't seem to apply to $f(x) = \frac1x$ with a = -1, b = 1 and y = 0, although the Wikipedia article states, that "[...] every continuous function is a Darboux function". Am i making an error or is there an "on an intervall" missing in the article? – Carlos Esparza Sep 07 '15 at 11:20
  • @0x539 the domain of $1/x$ is not connected, neither is the domain of $\tan$. – user251257 Sep 07 '15 at 11:38
  • @user251257 yet they are Darboux functions. The Wikipedia articles definition would not make them Darboux functions. – Carlos Esparza Sep 07 '15 at 11:46
  • @0x539 the intermediate value property makes only sense for functions with connected domain or which can be extended in that way. The wiki is sloppy – user251257 Sep 07 '15 at 11:56
  • @user251257 so any continuous function with a connected domain which satisfies the intermediate value property is a Darboux function. (or do they have to satisfy the intermediate property on a connected subset of their domain?) – Carlos Esparza Sep 07 '15 at 12:00
  • @0x539 continuous function on an interval (these are the only connected subsets of the reals) always has the intermediate value property. – user251257 Sep 07 '15 at 12:02
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This definition is not correct for functions whose domain is connected.

Darboux's theorem, also called the "intermediate value theorem for derivatives," states that all derivatives have your property. However, there are discontinuous derivatives, some of them wildly so.

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Rory Daulton
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