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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 strongler 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?

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?

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

$(*)$ for any real numbers $a<b$ with $f(a)\ne f(b)$ there exists a point $x\in I(a,b)$ such that $f(x)\in I[f(a),f(b)]$.

Here $I[a,b]$ and $I(a,b):=I[a,b]\setminus\{a,b\}$ stand for the closed and open segments with end-points $a,b$ (more precisely, $I[a,b]$ is the smallest connected subset of the real line, containing the points $a,b$).

Remark 1. Functions with a bit strongler property:

$(S)$ for any real numbers $a<b$ with $f(a)\ne f(b)$ there exists a continuity point $x\in I(a,b)$ of $f$ such that $f(x)\in I(f(a),f(b))$

are called Świątkowski functions.

Remark 2. Another stronger property

$(D)$ for any real numbers $a<b$ any any $y\in I(f(a),f(b))$ there exists $x\in I(a,b)$ with $f(x)=y$

is called the Darboux property.

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

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

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

But maybe this theorem is known? If yes, could you provide me with a suitable reference?

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 strongler 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?

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

$(*)$ for any real numbers $a<b$ with $f(a)\ne f(b)$ there exists a point $x\in I(a,b)$ such that $f(x)\in I[f(a),f(b)]$.

Here $I[a,b]$ and $I(a,b):=I[a,b]\setminus\{a,b\}$ stand for the closed and open segments with end-points $a,b$ (more precisely, $I[a,b]$ is the smallest connected subset containing the points $a,b$).

Remark 1. Functions with a bit strongler property:

$(S)$ for any real numbers $a<b$ with $f(a)\ne f(b)$ there exists a continuity point $x\in I(a,b)$ of $f$ such that $f(x)\in I(f(a),f(b))$

are called Świątkowski functions.

Remark 2. Another stronger property

$(D)$ for any real numbers $a<b$ any any $y\in I(f(a),f(b))$ there exists $x\in I(a,b)$ with $f(x)=y$

is called the Darboux property.

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

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

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

But maybe this theorem is known? If yes, could you provide me with a suitable reference?

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

$(*)$ for any real numbers $a<b$ with $f(a)\ne f(b)$ there exists a point $x\in I(a,b)$ such that $f(x)\in I[f(a),f(b)]$.

Here $I[a,b]$ and $I(a,b):=I[a,b]\setminus\{a,b\}$ stand for the closed and open segments with end-points $a,b$ (more precisely, $I[a,b]$ is the smallest connected subset of the real line, containing the points $a,b$).

Remark 1. Functions with a bit strongler property:

$(S)$ for any real numbers $a<b$ with $f(a)\ne f(b)$ there exists a continuity point $x\in I(a,b)$ of $f$ such that $f(x)\in I(f(a),f(b))$

are called Świątkowski functions.

Remark 2. Another stronger property

$(D)$ for any real numbers $a<b$ any any $y\in I(f(a),f(b))$ there exists $x\in I(a,b)$ with $f(x)=y$

is called the Darboux property.

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

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

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

But maybe this theorem is known? If yes, could you provide me with a suitable reference?