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?

Darboux function? $\endgroup$ – Francois Ziegler May 5 '18 at 12:12Darboux like functionsby 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