Taylor characterization of differentiability

An alternative definition of "differentiability" uses the -Taylor-
criterion:
a function is $k$ times differentiable -at a point- $a$ if it has a degree
$k$ taylor polynomial, IE, if there's a polynomial $P_k(x-a)$ s.t.
$f(x)-P_k(x-a) \in o((x-a)^k)$.
[and the derivatives are the coefficients of $\frac{1}{k!}x^k$]

Note that this is "the right" definition:
- for $k=0,1$ it's the definition of pointwise continuity and
  differentiability, resp.
  Indeed, this is The Right Way of thinking about differentiability
  [EG, in higher dimensions, you really need to think about the
   derivative as being a linear approximation]
- if $f$ is $k$ times differentiable in the traditional sense at $a$
  (IE, $(k-1)$-times differentiable on some nbhd of $a$
   and $f^{(k-1)}$ is differentiable at $a$)
  then by Taylor's theorem, it's differentiable in this sense.
- it's a pointwise definition!
  We like pointwise!

Does this make a difference?
Yes, of course.
Consider
$$f_k(x) := \begin{cases}
 x^k & \text{$k$ rational}\\
-x^k & \text{$k$ irrational}
\end{cases}$$
Then $f_k$ is $(k-1)$ times differentiable at $0$ in the Taylor sense,
but isn't even continuous anywhere other than $0$, so it can't be more
than once-differentiable (at $0$!) in the usual sense
b/c $f'$.

This POV is important enough, and clarifies
continuity/differentiability (at a point)
that it should be introduce -simultaneously- with the definition of
derivative!
This also underlines that (pointwise) differentiability is about
-polynomial asymptotics- (infintesimal polynomial approximation):
that is, that's -exactly- what it's about.
[this is a -tight- definition: no more nor less than needed.]
-----------------------------------------------
Incidentally, this suggests the use of a ``polynomial'' scale,
like a log scale, when graphing:
if the $y$ axis has equal markings for $1^3, 2^3, \dotsc$
then
$$f(x)=\begin{cases}
 x^3 & x\geq 0\\
-x^3 & x\leq 0
\end{cases}$$
will look like an absolute value: it'll clearly not be 3 times
differentiable.

[remember that the point of all these scales is that humans
 biologically can see lines but can't see exponentials (or cubics,
 or even quadratics too well)]
