I am taking an intro to real analysis course and a multi-variable calculus course simultaneously, so I am looking for a rigorous proof of the following.
We recently introduced continuity/differentiation/integration of a vector-valued function $\overrightarrow{r}(t)$, and learned that if each component in $\overrightarrow{r}(t)$ is continuous, then $\overrightarrow{r}(t)$ is continuous. This makes intuitive sense, since if each "direction" of $\overrightarrow{r}(t)$ is continuous over $t$, then $\overrightarrow{r}(t)$ has ought to be continuous, too. I feel as though we are taking this fact for granted and without much rigor. What would a proof for the continuity of $\overrightarrow{r}(t)$ look like? And what about a proof for the differentiability/integrability of $\overrightarrow{r}(t)$? Furthermore, how do we construct this proof to work in $\mathbb{R}^n$?