Let $n$ be a positive integer greater than 1. A function $F$ from a subset $T$ of $\mathbb{R}$ to the Cartesian product $\mathbb{R}^n$ is called a vector-valued function of one real variable. Such a function joins to any real number $t$ of $T$ a coordinate vector $$F(t) \;=\; (f_1(t),\,\ldots,\,f_n(t)).$$ Hence one may say that the vector-valued function $F$ is composed of $n$ real functions $t \mapsto f_i(t)$ , the values of which at $t$ are the components of $F(t)$ . Therefore the function $F$ itself may be written in the component form

(1)

Example. The ellipse $$\{(a\cos{t},\,b\sin{t})\,\vdots\;\; t \in \mathbb{R}\}$$ is the value set of a vector-valued function $\mathbb{R} \to \mathbb{R}^2$ ($t$ is the eccentric anomaly).

Limit, derivative and integral of the function (1) are defined componentwise through the equations

• $\displaystyle\lim_{t\to t_0}F(t) \;:=\; \left(\lim_{t\to t_0}f_1(t),\,\ldots,\,\lim_{t\to t_0}f_n(t)\right)$
• $\displaystyle F'(t) \;:=\; \left(f_1'(t),\,\ldots,\,f_n'(t)\right)$
• $\displaystyle\int_a^b\!F(t)\,dt \;:=\; \left(\int_a^b\!f_1(t)\,dt,\,\ldots,\,\int_a^b\!f_n(t)\,dt\right)$

Example. If $F$ is continuous on $[a,\,b]$ , the set

(2)

(3)