I will call a function $f$ almost smooth when for every point $a$ there are numbers $y_0$, $y_1$, $y_2$ ... $$f(x) = y_0 + \frac{y_1}{1!}(x-a) + \dots + \frac{y_n}{n!}(x-a) ^n + O((x-a)^{n+1})$$ for every natural $n$.
I am unsure that an almost smooth function may be not smooth.
My question: Do these "almost smooth" functions have any use in analysis? That is, it this concept useful? Can you provide some references on the theory of almost smooth functions?
First of all, $y_0=f(a)$. Secondly, almost smooth functions are differentiable. This is an immediate consequence of $$ f(x) = f(a) + y_1 (x-a) + o(x-a) . $$ In particular, $y_1=f'(a)$. Therefore, go and check your other proof because it is incorrect. They are not much better than that. In particular, the first derivative can be discontinuous and locally unbounded.
Moreover, almost smooth functions are closed under addition and multiplication (they form a ring), but most importantly they are also closed under composition. I claim that this is a Bad Thing™ and not a feature. The reason is basically that $(f\circ g)=(f'\circ g)g'$. When derivatives are bounded, the bounds are preserved, but when things get wild, this identity tells us that ill-behaviour gets amplified.
To get a taste of how horrible can these functions be, here are some examples. Notice that $$ f(x) = \exp\left(-\frac1{|x|}\right)g(x) $$ is almost smooth for every bounded function $g$ which is smooth in $\mathbb{R}\setminus\{0\}$. This is a large class of functions that can behave extremely bad near $0$, apart from decaying sufficiently fast. In particular, the first derivative can be unbounded. We can take for instance $$ f(x) = \exp\left(-\frac1{|x|}\right)\sin\left(\frac1{|x|}\right). $$ Here you can see a plot of such $f$. Doesn't look to bad; it's still quite nice... But now we employ the composition trick to produce a monster. I suggest you now to try to visualize $f\circ f$. Another example is $f(x+f(x))$. Here is a plot. Does it look "almost smooth"?
To conclude, I don't know of any good use of this class of functions, but i think that the name almost smooth doesn't represent well their true nature.