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?
