📂Vector Analysis

Limits and Continuity of Vector-Valued Functions

Definition¹

Let the vector function $\mathbf{r} : \mathbb{R} \to \mathbb{R}^{3}$ for three scalar functions $f, g, h: \mathbb{R} \to \mathbb{R}$ be given as follows. $$\mathbf{r}(t) = \left( f(t), g(t), h(t) \right)$$

Define the limit of $\mathbf{r}$ at $a$ as follows.

$$\lim\limits_{t \to a} \mathbf{r}(t) = \left( \lim\limits_{t \to a} f(t), \lim\limits_{t \to a} g(t), \lim\limits_{t \to a} h(t) \right)$$

We say that $\mathbf{r}$ is continuous at $a$ if the following equation holds.

$$\lim\limits_{t \to a} \mathbf{r}(t) = \mathbf{r}(a)$$

Explanation

This extends the definition of the limit and continuity of scalar functions directly. It is similarly defined for $n$ dimensions. For $\mathbf{r}(t) = \left( f_{1}(t), \dots, f_{n}(t) \right)$,

$$\lim\limits_{t \to a} \mathbf{r}(t) = \left( \lim\limits_{t \to a} f_{1}(t), \dots, \lim\limits_{t \to a} f_{n}(t) \right)$$