In the mathematical discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domains is a subset of some finite-dimensional Euclidean space. It is possible to generalize the notion of derivative to functions whose domain and codomain are subsets of arbitrary topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-valued function is a subset of a finite-dimensional Euclidean space then many of these notions become logically equivalent resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is also more well-behaved compared to the general case. This article presents the theory of ${\displaystyle k}$-times continuously differentiable functions on an open subset ${\displaystyle \Omega }$ of Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ (${\displaystyle 1\leq n<\infty }$), which is an important special case of differentiation between arbitrary TVSs. This importance stems partially from the fact that every finite-dimensional vector subspace of a Hausdorff topological vector space is TVS isomorphic to Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ so that, for example, this special case can be applied to any function whose domain is an arbitrary Hausdorff TVS by restricting it to finite-dimensional vector subspaces.

All vector spaces will be assumed to be over the field ${\displaystyle \mathbb {F} ,}$ where ${\displaystyle \mathbb {F} }$ is either the real numbers ${\displaystyle \mathbb {R} }$ or the complex numbers ${\displaystyle \mathbb {C} .}$

A map ${\displaystyle f,}$ which may also be denoted by ${\displaystyle f^{(0)},}$ between two topological spaces is said to be ${\displaystyle 0}$-times continuously differentiable or ${\displaystyle C^{0}}$ if it is continuous. A topological embedding may also be called a ${\displaystyle C^{0}}$-embedding.

Differentiable curves are an important special case of differentiable vector-valued (i.e. TVS-valued) functions which, in particular, are used in the definition of the Gateaux derivative. They are fundamental to the analysis of maps between two arbitrary topological vector spaces ${\displaystyle X\to Y}$ and so also to the analysis of TVS-valued maps from Euclidean spaces, which is the focus of this article.

A continuous map ${\displaystyle f:I\to X}$ from a subset ${\displaystyle I\subseteq \mathbb {R} }$ that is valued in a topological vector space ${\displaystyle X}$ is said to be (once or ${\displaystyle 1}$-time) differentiable if for all ${\displaystyle t\in I,}$ it is differentiable at ${\displaystyle t,}$ which by definition means the following limit in ${\displaystyle X}$ exists: $${\displaystyle f^{\prime }(t):=f^{(1)}(t):=\lim _{\stackrel {r\to t}{t\neq r\in I}}{\frac {f(r)-f(t)}{r-t}}=\lim _{\stackrel {h\to 0}{t\neq t+h\in I}}{\frac {f(t+h)-f(t)}{h}}}$$ where in order for this limit to even be well-defined, ${\displaystyle t}$ must be an accumulation point of ${\displaystyle I.}$ If ${\displaystyle f:I\to X}$ is differentiable then it is said to be continuously differentiable or ${\displaystyle C^{1}}$ if its derivative, which is the induced map ${\displaystyle f^{\prime }=f^{(1)}:I\to X,}$ is continuous. Using induction on ${\displaystyle 1<k\in \mathbb {N} ,}$ the map ${\displaystyle f:I\to X}$ is ${\displaystyle k}$-times continuously differentiable or ${\displaystyle C^{k}}$ if its ${\displaystyle k-1^{\text{th}}}$ derivative ${\displaystyle f^{(k-1)}:I\to X}$ is continuously differentiable, in which case the ${\displaystyle k^{\text{th}}}$-derivative of ${\displaystyle f}$ is the map ${\displaystyle f^{(k)}:=\left(f^{(k-1)}\right)^{\prime }:I\to X.}$ It is called smooth, ${\displaystyle C^{\infty },}$ or infinitely differentiable if it is ${\displaystyle k}$-times continuously differentiable for every integer ${\displaystyle k\in \mathbb {N} .}$ For ${\displaystyle k\in \mathbb {N} ,}$ it is called ${\displaystyle k}$-times differentiable if it is ${\displaystyle k-1}$-times continuous differentiable and ${\displaystyle f^{(k-1)}:I\to X}$ is differentiable.

A continuous function ${\displaystyle f:I\to X}$ from a non-empty and non-degenerate interval ${\displaystyle I\subseteq \mathbb {R} }$ into a topological space ${\displaystyle X}$ is called a curve or a ${\displaystyle C^{0}}$ curve in ${\displaystyle X.}$ A path in ${\displaystyle X}$ is a curve in ${\displaystyle X}$ whose domain is compact while an arc or C⁰-arc in ${\displaystyle X}$ is a path in ${\displaystyle X}$ that is also a topological embedding. For any ${\displaystyle k\in \{1,2,\ldots ,\infty \},}$ a curve ${\displaystyle f:I\to X}$ valued in a topological vector space ${\displaystyle X}$ is called a ${\displaystyle C^{k}}$-embedding if it is a topological embedding and a ${\displaystyle C^{k}}$ curve such that ${\displaystyle f^{\prime }(t)\neq 0}$ for every ${\displaystyle t\in I,}$ where it is called a ${\displaystyle C^{k}}$-arc if it is also a path (or equivalently, also a ${\displaystyle C^{0}}$-arc) in addition to being a ${\displaystyle C^{k}}$-embedding.

The definition given above for curves are now extended from functions valued defined on subsets of ${\displaystyle \mathbb {R} }$ to functions defined on open subsets of finite-dimensional Euclidean spaces.

Throughout, let ${\displaystyle \Omega }$ be an open subset of ${\displaystyle \mathbb {R} ^{n},}$ where ${\displaystyle n\geq 1}$ is an integer. Suppose ${\displaystyle t=\left(t_{1},\ldots ,t_{n}\right)\in \Omega }$ and ${\displaystyle f:\operatorname {domain} f\to Y}$ is a function such that ${\displaystyle t\in \operatorname {domain} f}$ with ${\displaystyle t}$ an accumulation point of ${\displaystyle \operatorname {domain} f.}$ Then ${\displaystyle f}$ is differentiable at ${\displaystyle t}$ if there exist ${\displaystyle n}$ vectors ${\displaystyle e_{1},\ldots ,e_{n}}$ in ${\displaystyle Y,}$ called the partial derivatives of ${\displaystyle f}$ at ${\displaystyle t}$, such that $${\displaystyle \lim _{\stackrel {p\to t}{t\neq p\in \operatorname {domain} f}}{\frac {f(p)-f(t)-\sum _{i=1}^{n}\left(p_{i}-t_{i}\right)e_{i}}{\|p-t\|_{2}}}=0{\text{ in }}Y}$$ where ${\displaystyle p=\left(p_{1},\ldots ,p_{n}\right).}$ If ${\displaystyle f}$ is differentiable at a point then it is continuous at that point. If ${\displaystyle f}$ is differentiable at every point in some subset ${\displaystyle S}$ of its domain then ${\displaystyle f}$ is said to be (once or ${\displaystyle 1}$-time) differentiable in ${\displaystyle S}$, where if the subset ${\displaystyle S}$ is not mentioned then this means that it is differentiable at every point in its domain. If ${\displaystyle f}$ is differentiable and if each of its partial derivatives is a continuous function then ${\displaystyle f}$ is said to be (once or ${\displaystyle 1}$-time) continuously differentiable or ${\displaystyle C^{1}.}$ For ${\displaystyle k\in \mathbb {N} ,}$ having defined what it means for a function ${\displaystyle f}$ to be ${\displaystyle C^{k}}$ (or ${\displaystyle k}$ times continuously differentiable), say that ${\displaystyle f}$ is ${\displaystyle k+1}$ times continuously differentiable or that ${\displaystyle f}$ is ${\displaystyle C^{k+1}}$ if ${\displaystyle f}$ is continuously differentiable and each of its partial derivatives is ${\displaystyle C^{k}.}$ Say that ${\displaystyle f}$ is ${\displaystyle C^{\infty },}$ smooth, ${\displaystyle C^{\infty },}$ or infinitely differentiable if ${\displaystyle f}$ is ${\displaystyle C^{k}}$ for all ${\displaystyle k=0,1,\ldots .}$ The support of a function ${\displaystyle f}$ is the closure (taken in its domain ${\displaystyle \operatorname {domain} f}$) of the set ${\displaystyle \{x\in \operatorname {domain} f:f(x)\neq 0\}.}$