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Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property:

$(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\in C$ with $f(a)< f(b)$ there exists a point $x\in C\setminus\{a,b\}$ such that $f(a)\le f(x)\le f(b)$.


Remark 1. Functions with a bit stronger property:

$(S)$ for any connected subset $C\subset \mathbb R$ and points $a,b\in C$ with $f(a)<f(b)$ there exists a continuity point $x\in C$ of $f$ such that $f(a)<f(x)<f(b)$

are called Świątkowski functions.

Remark 2. Another stronger property

$(D)$ for any connected subset $C\subset \mathbb R$ the image $f(C)$ is connected

is called the Darboux property.

So, functions with (*) can be called either weak Świątkowski function or weak Darboux functions. Are there any other names or ideas?


Motivation: I need to call somehow this property $(*)$ since I can prove a nice

Theorem 1. A function $f:\mathbb R\to\mathbb R$ is continuous if and only if it has closed graph and possesses the property $(*)$.

This theorem implies its own self-generalization:

Theorem 2. A function $f:X\to\mathbb R$ defined on first-countable path-connected topological space $X$ is continuous if and only if it has closed graph and possesses the property $(*)$.

But maybe these two theorems are known? If yes, could you provide me with a suitable reference?

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    $\begingroup$ A Darboux function? $\endgroup$ – Francois Ziegler May 5 '18 at 12:12
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    $\begingroup$ @FrancoisZiegler Darboux seems much stronger, it is a bit like the surjective version of what Taras Banakh is proposing. (oh I see this has been incorporated already) $\endgroup$ – Frank Waaldijk May 5 '18 at 12:17
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    $\begingroup$ If you haven't yet looked through the survey paper Darboux like functions by Richard G. Gibson and Tomasz Natkaniec [Real Analysis Exchange 22 #2 (1996-1997), pp. 492-533], then I recommend starting there, including its bibliography. After this, if you still haven't found anything, perhaps google the title of this paper for more recent papers. $\endgroup$ – Dave L Renfro May 5 '18 at 17:32
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    $\begingroup$ Yes, it suffices that $f$ is Darboux and $f^{-1}(\{r\})$ is closed for each rational $r$: W. Rudin, Principles of Mathematical Analysis, Chapter 4, exercise 19 $\endgroup$ – Dap May 6 '18 at 15:54
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    $\begingroup$ Maybe call it the 'discrete Swiatkowski property'; after all, it is the Swiatkowski property in the discrete topology... I generally feel that it is better to avoid generic adjectives (weak, strong, good, `quasi-' and so on) in terminology, because they are very much overloaded as is. $\endgroup$ – t3suji May 6 '18 at 17:04

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