Question

State the definition of continuity of a vector-valued function. Give an example of a vector-valued function that is defined but not continuous at \(t=2\).

Short answer

The continuity of a vector-valued function \(\mathbf{r}(t)\) at \(t=a\) demands that all its component functions be continuous at that point. As a case in point, the function \(\mathbf{r}(t)=\langle t, \frac{1}{t-2} \rangle\) is defined everywhere but is not continuous at \(t=2\).

Step-by-step solution

Definition of Continuity for a Vector-Valued Function

A vector-valued function \(\mathbf{r}(t)=\langle f(t),g(t),h(t) \rangle\) is said to be continuous at \(t=a\) if and only if each of its component functions \(f(t)\), \(g(t)\), and \(h(t)\) are continuous at \(t=a\). In other words, if \( \lim_{{t\to a}}f(t)=f(a)\), \(\lim_{{t\to a}}g(t)=g(a)\), and \(\lim_{{t\to a}}h(t)=h(a)\), the function is deemed continuous.

Non-continuous Vector-Valued Function Example

Now, let's consider the vector-valued function \(\mathbf{r}(t)=\langle t, \frac{1}{t-2} \rangle\). This function is defined everywhere except at \(t=2\). As the value approaches 2, the function's output heads toward infinity, meaning that the function is not continuous. Thus, \(\mathbf{r}(t)=\langle t, \frac{1}{t-2} \rangle\) is a vector-valued function defined but not continuous at \(t=2\).