Consider a linear map from $\Bbb R^n \rightarrow \Bbb R^m$.
Question 1. How can we show that all such linear maps are differentiable without loss of generality?
Question 2. Is it possible that even though a function is not continuous at a point, it can be differentiable at that point ? Say for example, consider a linear map $f: \Bbb R^2 \rightarrow \Bbb R$
$f(x,y) =
\begin{cases}
\frac{xy}{x^2 + y^2} & (x,y) \neq (0,0) \\[2ex]
0 & (x,y) = (0,0)
\end{cases}$
is not continuous at $(0,0)$ but is differentiable at $(0,0)$.
Best Answer
A linear map $\phi$ between finite dimensional spaces is always differentiable, and its derivative at a given point is given by $\phi$.
This is immediate from the definition, since we have an identity
$$\phi(a+h) = \phi(a) + \phi(h)$$
so there is not even a lower order term.
Differentiability in a point $p$ implies continuity in that point.