Question states: "Show that a function $f: \Bbb R^n \to \Bbb R^m$ is differentiable at $a$ if and only if all of its components $f_j: \Bbb R^n \to \Bbb R$ are differentiable for $1 \le j \le m$."
My instructor requires the use of epsilon delta proof for this problem. Although the concept in this problem is very intuitive, I couldn't really get a grip on this. I have been fiddling with the Jacobian matrix for couple hours, but I am really stuck. Any help would be great. I really want to learn how this can be proven rigorously.
We can define the functions $\pi_j:\mathbb{R^m}\to\mathbb{R},\pi_j(x_1,x_2,\ldots,x_m)=x_j$ and $g_j:\mathbb{R}\to\mathbb{R^m}, g_j(x)=(0,0,\ldots,x,\ldots,0)$ where the $x$ is at the $j$-th position. We can check that they are linear functions, hence differentiable. Now we see that $f_j=\pi_j\circ f$ and $\displaystyle f=\sum_{i=1}^m g_j\circ f_j$. Hence the statement follows.