What is this notion of differentiability?

Question

The standard definition for the derivative of a function $f: R^n \to R^m$ at a point $x_0$ is the unique linear map, $f'(x_0): R^n \to R^m$, such that $$ \lim_{x \to x_o}\frac{\|f(x) - f(x_0) - f'(x_0)(x - x_0)\|}{\|x - x_0\|} = 0$$

On the other hand, the brain-dead generalization of the single-variable definition might be something like this: $$f'_{stupid}(x_0) = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{\|x - x_0\|}$$

which seems to say something like "no matter what direction I take to approach $x_0$, the "$m$-d slope'' of $f$ is the same and is equal to $f'_{stupid}(x_0$)." Or, on other words, that all directional derivatives are equal to $f'_{stupid}(x_0)$.

Can anyone shed some light on any relations between the two?

Answer 1

One relation between these two definitions is that if the first definition holds then the limit in the braid-dead definition does not even exist (outside of a very exceptional case).

The thing to keep in mind is that in order for a multidimensional limit $\lim_{x \to x_0}$ to exist, the value of the limit must be the same no matter how $x$ approaches $x_0$. This even holds in the 1-dimensional case of course: we require the left and right limits to be equal, in order for the limit to exist.

So, let's suppose that the first (correct) definition holds, and consider the linear transformation $f'(x_0) : R^n \to R^m$. Suppose that $f'(x_0)$ is not the zero transformation, in other words there exists a unit vector $u \in R^n$ such that the image vector $w = f'(x_0)(u)$ is not the zero vector. Then the limit that defines $f'_{stupid}(x_0)$ does not exist. To prove this, first let $x$ approach $x_0$ along the ray $x_0 + t u$ for $t > 0$ you get a certain nonzero value which is $$\lim_{t \to 0^+} \frac{f(x_0+tu) - f(x_0)}{\| t u \|} = f'(x_0)(u) = w $$ whereas if you let $x$ approach $x_0$ along the ray $x_0 + t \vec u$ for $t < 0$ then you get the opposite value $$\lim_{t \to 0^-} \frac{f(x_0+tu) - f(x_0)}{\| t u \|} = f'(x_0)(-u) = -w $$ The exceptional case is that $f'(x_0)$ is the zero transformation, in which case all directional deriatives are zero, the limit in the brain dead definition exists, and its value is zero.