Continuity of a vector-valued function

A vector-valued function is continuous exactly where each of its three component functions is.

A limit of a vector-valued function is checked componentwise: $\lim_{t\to a}\mathbf{r}(t) = \left\langle \lim_{t\to a}f(t),\ \lim_{t\to a}g(t),\ \lim_{t\to a}h(t)\right\rangle$ , provided all three component limits exist. $\mathbf{r}(t)$ is continuous at $t=a$ exactly when $\lim_{t\to a}\mathbf{r}(t) = \mathbf{r}(a)$ , which happens exactly when $f$ , $g$ , and $h$ are each individually continuous at $a$ — one broken component is enough to break the whole curve, even if the other two behave perfectly.

Examples

$\mathbf{r}(t) = \left\langle \dfrac{t^2-4}{t-2},\ \sqrt{t+1},\ e^t\right\rangle$ is continuous on $[-1,2)\cup(2,\infty)$ , discontinuous only at $t=2$ .