I have a few questions:
If the directional derivatives $D_uf(a)$ exist for all directions $u$ and depend linearly on $u$, must $f$ be differentiable at $a$?
Also, how do I show that the function
$f: \mathbb R^3 \to \mathbb R^2$
$f(x, y, z) = (e^{x+y+z},$cos x2y) is everywhere differentiable without making use of partial
derivatives?
Hint:
For the first part, what does it mean if the directional derivative exists in every direction? What about the linear aspect?
For the second part of your question, are the exponential function and the cosine function differentiable? What about if you compose a differentiable function with another, is it still differentiable?