\documentclass{Cauchy} \title{A Review of Pompeiu-Hausdorff Metric Differentiability and Its Relation to Generalized Hukuhara Differentiability \\ } \newcommand{\shorttitle}{A Review of Pompeiu-Hausdorff Metric Differentiability and Its Relation to Generalized Hukuhara Differentiability} \newcommand{\authemail}{mslk@ub.ac.id} %Only the corresponding author's email \newcommand{\AuthorOne}{William Surya Putra,Mohamad Muslikh} % === Uncomment ONE of the two blocks below === % ===== CASE 1: Shared affiliation ===== % \author{First Author} % \author{Second Author\CorrAuth} % \author{Third Author} % \author{Fourth Author} % \affil{Affiliation of The Authors} % ===== CASE 2: Different affiliations ===== \author[1]{William Surya Putra} \author[2\CorrAuth]{Mohamad Muslikh} \affil[1,2]{Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Brawijaya, Indonesia.} \begin{document} \label{firstpage} \maketitle \begin{abstract}\noindent The Pompeiu-Hausdorff distance/Pompeiu-Hausdorff metric is a concept in analysis that measures the distance between two subsets of a metric space, one of its important applications being the Hausdorff metric differentiability of set-valued functions . This article reviews the definition and properties of Pompeiu-Hausdorff distance differentiability on the space of compact and convex subsets of the Euclidean space with dimension n. We also present concepts about the generalized Hukuhara difference and its differentiability. By studying both topics, we discuss the established relationship between Pompeiu-Hausdorff metric differentiability and generalized Hukuhara differentiability. \end{abstract} \begin{keywords} Generalized Hukuhara difference,Hukuhara difference ,Pompeiu-Hausdorff metric, Set-valued function \end{keywords} \CopyRight \section{Introduction} Set-valued analysis is one of the topics in mathematical analysis that studies collections of sets \cite{aubin2009} . This field has a wide range of applications, such as in control theory, economics, and artificial intelligence \cite{aubin2009},\cite{aubin2000}. An important topic in set-valued analysis is the differentiability of set-valued functions, particularly Hausdorff distance differentiability in metric spaces which has application such as in pattern analysis for comparing image \cite{huttenlocher1993}. Another important topic is the study of Hukuhara difference differentiability of set-valued functions. One such instance is the use in defining the derivative of Fuzzy set-valued mappings \cite{aliev2018} . The generalized Hukuhara difference differentiability is the generalization of Hukuhara differentiability concept usually used in development of differential calculus for interval-valued functions \cite{chalco2013}. Recently, the notion has also been used in the study of metric derivatives for set-valued functions to define the differentiation of set-valued function and is used in proving the differentiation properties of Lipschitz set-valued function \cite{muslikh2019}. The concept of set-valued analysis was introduced by Aubin \cite{hukuhara}. Later, Masuo Hukuhara developed the notion of the Hukuhara difference for compact and convex sets in order to define Hukuhara differentiability of set-valued mappings \cite{hukuhara}. This theory was later extended by Stefanini and Bede, who introduced the generalized Hukuhara difference and generalized Hukuhara differentiability of set-valued mappings \cite{stefanini2009}. More recently, Khastan, López, and Shahidi established the differentiability of set-valued mappings under the Pompeiu-Hausdorff metric or known as Pompeiu-Hausdorff distance on the space of compact and convex subsets of the Euclidean space and studied its properties \cite{khastana2021}. However, in the existing literature, several fundamental proofs such as the necessary and sufficient condition for the existence of the Hukuhara difference and fundamentals properties of Pompeiu-Hausdorff metric differentiability are often not given. In this paper we review definition and provides proof of fundamentals properties for the mentioned topics above. After this we will discuss the established relationship between Pompeiu-Hausdorff differentiability and generalized Hukuhara differentiability. \section{Preliminaries.} \label{sec:preliminaries} In this section, we provide the definitions that will be used later on results and discussion part. \subsection{Set-Valued Mappings } \begin{definition}[set-valued mapping, \cite{muslikh2022},\cite{rockafellar1998},\cite{rafalgoebel2024},\cite{aubin1984}] \\ Let $F \colon X \to \mathcal{P}(Y)$. The mapping $F$ is called a \textit{set-valued function} if \[ F(x) \neq \emptyset \quad \] for every $x \in X$. \end{definition} \begin{example} Define $F \colon [0,1] \to \mathcal{P}(\mathbb{R})$ by \[ F(x) = [0,x] \quad \text{for } x \in [0,1]. \] Then $F$ is a set-valued function. \end{example} \subsection{Minkowski sum, and Minkowski scalar multiplication} \begin{definition}[minkowski sum and minkowski scalar multiplication, \cite{muslikh2022},\cite{schneider2013}]\\ Let $A$ and $B$ be subsets of euclidean space $\mathbb{R}^k$ and $\alpha \in \mathbb{R}$. Denote \[ A + B = \{a + b \mid a \in A,\, b \in B \}, \qquad \alpha A = \{ \alpha a \mid a \in A \}. \tag{2.9} \] This definition is known as \textit{Minkowski sum} and \textit{Minkowski scalar multiplication} respectively. \end{definition} \begin{example} Let $A=[0,3]$ and $B=[2,5]$ , then the Minkowski sum between $A$ and $B$ is $C=[0+2,3+5]=[2,8]$,and the Minkowski scalar multiplication between $A$ and $k\in \mathbb{R}^{+}$ is $kA=[0,3k]$ \end{example} \subsection{Real linear space, normed linear space } \begin{definition}[real linear space, \cite{aram2016},\cite{sheldon2023}] Recall that a real linear space $X$ is a commutative group with respect to addition, and for each element (called a vector) $x \in X$ and each real number $\alpha \in \mathbb{R}$, the scalar multiplication $\alpha x \in X$ is defined. The operations of addition and scalar multiplication satisfy the following axioms for all $\alpha, \beta \in \mathbb{R}$ and $x, y \in X$: \begin{enumerate} \item $ \alpha(x + y) = \alpha x + \alpha y; $ \item $ (\alpha + \beta)x = \alpha x + \beta x; $ \item $ (\alpha \beta)x = \alpha (\beta x); $ \item $ 1 \cdot x = x. $ \end{enumerate} \end{definition} \begin{definition}[Normed Linear Space, \cite{aram2016},\cite{mahlon1958},\cite{sen2013},\cite{giles2012}] A real linear space $X$ is called a \textit{normed linear space} if it is equipped with a function $\|\cdot\| : X \to \mathbb{R}$, called a norm, satisfying the following axioms for all $x, y \in X$ and all $\alpha \in \mathbb{R}$: \begin{enumerate} \item $ \|x\| \ge 0 \quad \text{and} \quad \|x\| = 0 \iff x = 0; $ \item $ \|\alpha x\| = |\alpha| \, \|x\|; $ \item $ \|x + y\| \le \|x\| + \|y\|. $ \end{enumerate} \end{definition} \subsection{Convex sets} \begin{definition}[Convex sets, \cite{aram2016},\cite{jean2001},\cite{boyd2004},\cite{barvinok2002},\cite{edwin2008}]\\ Let $X$ be a real linear space and let $A \subseteq X$ The set $A$ is called \textit{convex} if for any two points $x_1, x_2 \in A$, the line segment between them is contained in $A$, that is, \[ [x_1, x_2] \subseteq A \quad \text{where} \quad [x_1, x_2] = \{\, \alpha x_1 + (1-\alpha)x_2 \mid \alpha \in [0,1] \,\}. \] Equivalently, \[ \alpha x_1 + (1-\alpha)x_2 \in A \quad \text{for all } \alpha \in [0,1]. \] \end{definition} \begin{example} The interval $[a,b]$, where $a,b \in \mathbb{R}$, is an example of a convex set, as any point between $a$ and $b$ lies inside the interval. In a sense, any line segment connecting two points inside the interval is contained in it. \end{example} \begin{definition}[Convex hull of a set, \cite{aram2016}]\\ Let $X$ be a real linear space and let $A \subseteq X$ . Denote convex hull of $A$ , $conv(A)=\bigcap_{C\in Y}C$ where $Y=\{C|C\subseteq X, A\subseteq C, C \space\text{convex set} \}$ as the smallest set containing $A$ that is also a convex set \end{definition} \begin{example} The convex hull of $\{0,1\}$ on $\mathbb{R}$ is the interval $[0,1]$ alternatively, we write $conv(\{0,1\})=[0,1]$ \end{example} \begin{properties} \[ \forall A,B \subseteq X \rightarrow conv(A+B)=conv(A)+conv(B)\] \end{properties} \subsection{Metric Space} \begin{definition}(Metric Space,\cite{copson1968}) A \emph{metric space} is a set $X$ equipped with a distance function \[ d : X \times X \to \mathbb{R}, \] that assigns $d(x,y)$ to each pair of points $x,y \in X$, and satisfies the following properties for all $x,y,z \in X$: \begin{enumerate} \item $ d(x,y) \ge 0$ \item $d(x,y) = 0 \iff x = y$ \item $d(x,y) = d(y,x)$ \item $d(x,z) \le d(x,y) + d(y,z)$ \end{enumerate} \end{definition} \subsection{Pompeiu--Hausdorff Distance} \begin{definition}(Pompeiu--Hausdorff Distance, \cite{khastana2021},\cite{rucklidge1996}, \cite{huttenlocher1993})\\ The Pompeiu--Hausdorff distance $H$ on the normed space $\mathcal{K}_C(\mathbb{R}^n)$ is defined by \[ H(A,B) = \max \left\{ \sup_{a \in A} d(a,B), \; \sup_{b \in B} d(b,A) \right\}, \quad A, B \in \mathcal{K}_C(\mathbb{R}^n), \] where \[ d(a,B) = \inf_{b \in B} \| a - b \|, \qquad d(b,A) = \inf_{a \in A} \| b - a \|. \] Here, $\|\cdot\|$ denotes the Euclidean norm on $\mathbb{R}^n$.It is known that $\left(\mathcal{K}_C(\mathbb{R}^n), H\right)$ is a complete metric space.Where $\mathcal{K}_C(\mathbb{R}^n)$ is the family of compact convex subset of $\mathbb{R}^n$ \end{definition} \begin{example} For $A=[0,1]$ and $B=[5,7]$ $H(A,B)=6$ since $\sup_{a \in A} d(a,B)= \sup_{a \in A}\inf_{b \in B} \| a - b \|= \sup_{a \in A}\| a - 5 \|=5$ and $\sup_{b \in B} d(b,A)=\sup_{b \in B} \inf_{a \in A} \| b - a \|=\sup_{b \in B}\|b-1\|=6$ \end{example} \subsection{Hukuhara Difference, Generalized Hukuhara Difference} \begin{definition}(Hukuhara Difference, \cite{phil1994})\\ Let $A, B \in \mathcal{K}_C(\mathbb{R}^n)$, where $\mathcal{K}_C(\mathbb{R}^n)$ denotes the family of all non-empty compact and convex subsets of $\mathbb{R}^n$. The \textit{ Hukuhara difference} of $A$ and $B$ is a set $C$ such that \[ A \,-_{\mathrm{H}}\, B = C \quad \Longleftrightarrow \quad A = B + C \ \] \end{definition} \begin{example} For $A=[0,1]$ and $B=[5,7]$ ,$B\,-_{\mathrm{H}}\, A = [5,6]$ since $[0,1]+[5,6]=[5,7]$ , however the converse $A\,-_{\mathrm{H}}\, B $ does not exist since the length of the intervals denoted by $len(I)$, $len(B)=2>1=len(A)$. \end{example} \begin{definition}(Generalized Hukuhara Difference, \cite{stefanini2009},\cite{stefanini2008})\\ Let $A, B \in \mathcal{K}_C(\mathbb{R}^n)$, where $\mathcal{K}_C(\mathbb{R}^n)$ denotes the family of all non-empty compact and convex subsets of $\mathbb{R}^n$. The \textit{generalized Hukuhara difference} of $A$ and $B$ is a set $C$ such that \[ A \,-_{\mathrm{gH}}\, B = C \quad \Longleftrightarrow \quad A = B + C \ \text{or} \ B = A + (-C) \] \end{definition} \begin{example} For $A=[7,11]$ and $B=[-3,10]$ ,$A\,-_{\mathrm{gH}}\, B = [1,10]$ since $[7,11]-[1,10]=[-3,10]$. Notice that in this case $A\,-_{\mathrm{gH}}\, B$ exist but $A\,-_{\mathrm{H}}\, B$ does not,due to $len(A)=4=0, \forall A \in KC(\mathbb{R}^k)$: Clearly $(a+b)A \subseteq aA+bA$ we shall prove the converse. take any $x \in aA , y\in bA $ .Write $x=aw,y=bt, w,t\in A$ Notice that $(a+b)(\frac{x}{a+b} +\frac{y}{a+b})=(a+b)(\frac{a}{a+b}w +\frac{b}{a+b}t)$ clearly $\frac{a}{a+b}w +\frac{b}{a+b}t\in A$ hence $x+y \in (a+b)A$ \end{proof} \end{proposition} \subsection{Necessary and Sufficient condition for Hukuhara and Generalized Hukuhara difference existence} \begin{proposition}Let $U, V \in \mathcal{K}_c(\mathbb{R}^k)$. The necessary and sufficient condition for the Hukuhara difference $U -_HV$ to exist is that for all $u \in U$ there exists at least a point $w$ such that $$ u \in V + \{w\} \subseteq U.$$ \begin{proof} Necessary condition proof: Since $U-_H V$ then $\exists W$ such that $U=W+V$ hence $\forall u\in U, u=v+w, v\in V, w\in W$. From here we get $u\in V+\{w\} \subseteq W+V=U$\\ Sufficient condition proof : Pick any $u \in U$. $\exists w_u \in \mathbb{R}^n$ so that $$ u \in V + \{w_u\} \subseteq U. $$ We get $$ U \subseteq \bigcup_{u \in U} (V + \{w_u\}) = V + \bigcup_{u \in U} \{w_u\} = V+W \subseteq U $$ Hence $$ U = V+W. $$\\ Denote $cl(U)$ as closure of $U$ and $conv(U)$ as the smallest convex sets containing $U$.\\ We shall prove $$U=V+cl(W), U=V+conv(W)$$, $U=V+W\subseteq cl(V)+cl(W)=V+cl(W)\subseteq cl(V+W)=cl(U)=U$ and $U=V+W\subseteq conv(V)+conv(W)=V+conv(W)\subseteq conv(V+W)=conv(U)=U$ .Hence $cl(conv(W))=Q$ so $Q$ is convex and $U=V+Q$. Since $U,V$ are bounded then $Q$ is bounded too. Hence $Q\in KC(\mathbb{R}^n)$. $Q$ unique , if not then $\exists B\in KC(\mathbb{R}^n)$ so that $U=V+B$ hence $0=H(V+Q,V+B)=H(Q,B) \rightarrow Q =B$.Hence $U-_H V=Q$. \end{proof} \end{proposition} \begin{example}(Intuition) Consider 2 closed circles on cartesian coordinate centered at its origin with radius respectively 2 and 1. Denote the bigger circle as $A$ and the smaller circle $B$. Notice that by translating circle $B$ by any points from itself will forms a translated circle $B$ that is inside $A$. Taking the union of every possible translated circle $B$ will gives us the entire circle $A$ . Hence $A-_HB=B$ \end{example} \begin{proposition}Let $U, V \in \mathcal{K}_c(\mathbb{R}^k)$. The necessary and sufficient condition for the generalized Hukuhara difference $U -_{gH}V$ to exist is that for all $v \in V$ and $u \in U$ there exists at least a point $w$ such that either $$ v \in U - \{w\} \subseteq V$$ or $$ u \in V + \{w\} \subseteq U$$ \begin{proof} By noticing that if $ W \in \mathcal{K}_c(\mathbb{R}^k)$ then $ -W \in \mathcal{K}_c(\mathbb{R}^k)$ where $-W=(-1)W$ from Minkowski scalar multiplication definition , the proof follows the same logic and argument as in Hukuhara differences case. \end{proof} \end{proposition} \subsection{Basic Properties in Pompeiu-Hausdorff differentiation} \begin{proposition} The value \( A^{+} \) is unique. \begin{proof} \textbf{Case 1:} Let there exist other sets \( B^{+} \) so that both \( B^{+} \) and \( A^{+} \) satisfy (i) in definition 14. Then from the definition, pick \( p > 0 \) such that \[ \frac{H(F(t+h), F(t) + hA^{+})}{h} < s, \quad \forall s > 0, \] and for any \( h,\, 0 < h < c \), \[ \frac{H(F(t+h), F(t) + hB^{+})}{h} < y, \quad \forall y > 0. \] Pick \( r = \min(c,p) \) so that \( \forall h \) satisfying \( 0 < h < r \) we have: \[ \frac{H(F(t) + hA^{+}, F(t) + hB^{+})}{h} < y + s. \] Hence, \[ \frac{H(F(t) + hA^{+}, F(t) + hB^{+})}{h} = H(A^{+}, B^{+}) < y + s \Longrightarrow A^{+} = B^{+}. \] \textbf{Case 2:} Let \( B^{+} \) follow (i) and \( A^{+} \) follow (ii) in definition 14. Then \( \forall h > 0 \) we have \[ 0 \leq H(0, -A^{+} + B^{+}) = \frac{H(F(t+h) + F(t), F(t+h) + F(t) - hA^{+} + hB^{+})}{h} \] \[ \leq \frac{H(F(t), F(t+h) - hA^{+})}{h} + \frac{H(F(t+h), F(t) + hB^{+})}{h} \tag{iii} \] Consider \( h \to 0^{+} \) ,then RHS \( = 0 \). According to squeeze theorem, \[ H(\{0\}, B^{+} - A^{+}) = 0 \quad \Rightarrow \quad B^{+} = A^{+}. \] \end{proof} \end{proposition} \begin{proposition} \( F' = \{0\} \) , if \( F = B \) (i.e., \( F \) is constant). \begin{proof} If $F$ is constant then $F(t)=F(t+h)$ ,hence \[ \lim_{h \to 0^{+}} \frac{1}{h} H(F(t+h), F(t) ) =\lim_{h \to 0^{+}} H(\{0\}, \{0\} )= 0, \lim_{h \to 0^{-}} \frac{1}{h} H(F(t+h), F(t) ) =\lim_{h \to 0^{-}} H(\{0\}, \{0\})= 0, \] which implies $F'(t)=\{0\}$ \end{proof} \end{proposition} \begin{proposition} If \( F \) is differentiable at the point \( x=t \), then \( F \) is continuous at \( x=t \). \begin{proof} Since the proof for the other case follows the same logic ,we prove one case only. Now assume it satisfy (1) in definition 15 .Since $F'(t)$ follows (1),then \[0<=H(F(t+h),F(t))=H(F(t+h)+hF'(t),F(t)+hF'(t))<=H(F(t+h),F(t)+hF'(t))+H(hF'(t),0)\] by limiting $h->0^{+}$ we have $H(F(t+h),F(t)+hF'(t))=0$ and $H(hF'(t),0)$ so $RHS=0$ by squeeze theorem $lim_{h \to 0^{+}} H(F(t+h),F(t))=0 $ using the same logic \[0<=H(F(t+h),F(t))=H(F(t+h)+hF'(t),F(t)+hF'(t))<=H(F(t+h),F(t)+hF'(t))+H(hF'(t),0)=0\] for $h->0^{-}$ so $lim_{h \to 0^{-}} H(F(t+h),F(t))=0 $ therefore $lim_{h \to 0} H(F(t+h),F(t))=0 $ which implies $F$ is continous at $x=t$ \end{proof} \end{proposition} \begin{proposition} The set of all Hausdorff$ differentiable$ set-valued functions that satisfy the same condition among (1), (2), (3), or (4) in definition 15 forms a linear space. \begin{proof} Let the set be denoted by $\mathcal{A}$. Pick two arbitrary elements of $\mathcal{A}$, $F$ and $G$, and without loss of generality assume $F,G$ satisfy (1). \medskip \textbf{1. $kF' = (kF)'$.} Let $F$ be Pompeiu--Hausdorff distance differentiable and satisfy (1). Let $kF = W$. Then: \begin{align} \lim_{h \to 0^{+}} \frac{1}{h} H(F(t+h), F(t) + hF'(t)) &= 0. \label{eq:kFforward} \end{align} On the other hand: \begin{align} \lim_{h \to 0^{+}} \frac{|k|}{h} H(F(t+h), F(t) + hF'(t)) &=\lim_{h \to 0^{+}} \frac{1}{h} H(kF(t+h), kF(t) + hkF'(t)) =0. \label{eq:kFforward} \end{align} But, \begin{align} \lim_{h \to 0^{+}} \frac{1}{h} H(kF(t+h), kF(t) + hkF'(t)) &=\lim_{h \to 0^{+}} \frac{1}{h} H(W(t+h), W(t) + hkF'(t)) \end{align} Hence \[ kF' = (kF)'. \] \medskip \textbf{2. $F' + G' = (F + G)'$.} Let $Q(t) = F(t) + G(t)$,$W=\frac{H(F(t+h),\, F(t) + hF'(t))}{h} + \frac{H(G(t+h),\, G(t) + hG'(t))}{h}$, and\\ $P= \frac{H(F(t),\, F(t-h) + hF'(t))}{h} + \frac{H(G(t),\, G(t-h) + hG'(t))}{h} $ . So \begin{align} 0 \leq \frac{H(F(t+h) + G(t+h),\, F(t) + G(t) + h(F'(t) + G'(t)))}{h} &\leq W . \label{eq:sumineq} \end{align} Taking $h \to 0^{+}$, \begin{align} 0 &\leq \lim_{h \to 0^{+}} \frac{H(F(t+h) + G(t+h),\, F(t) + G(t) + h(F'(t) + G'(t)))}{h} \leq \lim_{h \to 0^{+}} W = 0. \label{eq:sumlimit} \end{align} Thus, \begin{align} \lim_{h \to 0^{+}} \frac{H(F(t+h) + G(t+h),\, F(t) + G(t) + h(F'(t) + G'(t)))}{h} = \lim_{h \to 0^{+}} \frac{H(Q(t+h),\, Q(t) + hQ'(t))}{h} = 0. \label{eq:forwardQ} \end{align} Similarly, \begin{align} 0 \leq \frac{H(F(t) + G(t),\, F(t-h) + G(t-h) + h(F'(t) + G'(t)))}{h} &\leq P. \label{eq:sumineq-shifted} \end{align} Taking $h \to 0^{+}$, \begin{align} 0 &\leq \lim_{h \to 0^{+}} \frac{H(F(t) + G(t),\, F(t-h) + G(t-h) + h(F'(t) + G'(t)))}{h}\leq \lim_{h \to 0^{+}} P = 0. \label{eq:sumlimit-shifted} \end{align} Therefore, \begin{align} \lim_{h \to 0^{+}} \frac{H(F(t) + G(t),\, F(t-h) + G(t-h) + h(F'(t) + G'(t)))}{h} = \lim_{h \to 0^{+}} \frac{H(Q(t),\, Q(t-h) + hQ'(t))}{h} = 0, \end{align} The prove for the other case follows the same logic Hence \[ (F + G)' = F' + G'. \] \end{proof} \end{proposition} \subsection{Generalized Hukuhara differentiability and Pompeiu-Hausdorff distance differentiation relation} \begin{proposition} $F$ is $gH-differentiable$ iff $F$ is Pompeiu-Hausdorff\space$ differentiable$ \begin{proof} ($\Rightarrow$) Let $F$ be $gH-differentiable$. \quad Denote $F(t+x)-_{gh} F(t) =Q(x), x>0$ and $F'_{gh}(t)= \lim_{x \to 0^{+}} \frac{Q(x)}{x}$. Therefore, \[\lim_{x \to 0^{+}} \frac{1}{x} H(F(t+x), F(t) + x(F'_{gh}(t)) =\lim_{x \to 0^{+}} \frac{1}{x} H(F(t+x) +(-1)Q(x), F(t)+(-1)Q(x) + x(F'_{gh}(t))\] Hence, \[\lim_{x \to 0^{+}} \frac{1}{x} H(F(t),F(t)+(-1)Q(x)+ x(F'_{gh}(t))=\lim_{x \to 0^{+}} \frac{1}{x} H({0}, (-1)Q(x)+ x(F'_{gh}(t))\] Finally ,\[\lim_{x \to 0^{+}} \frac{1}{x} H({0}, (-1)Q(x)+ x(F'_{gh}(t))=\lim_{x \to 0^{+}} H(\{0\},\frac{(-1)Q(x)}{x}+F'_{gh}(t)) =0\] .The left differentiable case follows the same flow \medskip ($\Leftarrow$) Now assume $F$ is {Pompeiu-Hausdorff }\space$differentiable$. \quad We shall divide $F$ into 2 different cases\\ \textbf{Case 1:} $F$ follows (1) in definition 15. Since, \[\lim_{h \to 0^{+}} \frac{1}{h} H(F(t+h), F(t) + hF'(t)) = 0\] Then , \[\lim_{h \to 0^{+}} H(F(t+h), F(t) + hF'(t)) = 0\] So for sufficiently small $h->0^{+}$ , \[F(t+h)=F(t) + hF'(t)\] By $gH$ difference definition, \[F(t+h)-_{gH}F(t)= hF'(t)\] Therefore, \[F'(t)=\lim_{h \to 0^{+}}\frac{F(t+h)-_{gH}F(t)}{h}\tag{a}\] On the other hand, \[ \lim_{h \to 0^{+}} \frac{1}{h} H(F(t), F(t-h) + hF'(t)) = 0\] Hence, \[ \lim_{h \to 0^{+}} H(F(t), F(t-h) + hF'(t)) = 0\] For sufficiently small $h->0^{+}$ , \[F(t)=F(t-h) + hF'(t)\] Now assume $u=-h$, \[F(t)=F(t+u) -uF'(t)\] Finally according to definition, \[uF'(t)=F(t+u)-_{gH}F(t)\] \[F'(t)=\lim_{u \to 0^{-}}\frac{F(t+u)-_{gH}F(t)}{u}\tag{b}\] Hence by $(a)\space \text{and} \space (b)$ we deduce that $F$ is $gH-$ differentiable.\\ \textbf{Case 2:} $F$ follows (3) in definition 15. From (3) we have, \[\lim_{h \to 0^{-}} \frac{1}{h} H(F(t+h), F(t) + hF'(t)) = 0\] Hence, \[\lim_{h \to 0^{-}} H(F(t+h), F(t) + hF'(t)) = 0\] For sufficiently small $h->0^{-}$ \[ F(t+h)= F(t) + hF'(t)\] Therefore by $gH-$\space definition , \[hF'(t)=F(t+h)-_{gH}F(t)\] Finally, \[F'(t)=\lim_{h \to 0^{-}}\frac{F(t+h)-_{gH}F(t)}{h}\] Hence $F$ is $gH$ differentiable.\\ \textbf{The proof for the other cases follows the same logic and arguments as in definition (1) and (3)} \end{proof} \end{proposition} \section{Conclusion} \label{sec:conclusion} We have reviewed the fundamentals properties in Pompeiu-Hausdorff distance differentiability and concepts in generalized Hukuhara differentiability. Any set-valued function is gH differentiable iff it is Pompeiu-Hausdorff distance differentiable . The collection of every sets that is Pompeiu-Hausdorff distance differentiable that satisfy the same definition of differentiable forms linear space. The Pompeiu-Hausdorff distance behaves the same way as single-valued function like interval calculus in a sense that every set valued function that is Pompeiu-Hausdorff distance differentiable on a particular points implies continuity on that points. The hukuhara differences is unique and does not always exist , the same goes for generalized version of it in terms of its existence. \section*{CRediT Authorship Contribution Statement} \textbf{William Surya Putra:} Writer of this article . \\ \textbf{Mohamad Muslikh:} Advisor, and reviewer for this article. \section*{Declaration of Generative AI and AI-assisted technologies} ChatGPT was used in preparing this manuscript to assist in refining the grammar and making the mathematical expressions tidier and more readable. Every mathematical aspect was written and developed by the author. \section*{Declaration of Competing Interest} The author declares no competing interests. \section*{Funding and Acknowledgments} This article was funded by the Faculty of Mathematics, Brawijaya University, under the MBKM–B scheme, as part of a research assistantship supervised by Mohammad Muslikh, based on Document No. 06455.12/UN.10.F0901/B/PT/2025. \printbibliography \end{document}