Let $f: \mathbb{R} \rightarrow \mathbb{R^m}$ and $N: \mathbb{R} \rightarrow \mathbb{R^m}$ be two mappings satisfying:
$$\lim_{h\to0} \frac{f(a+h)-f(a)-N(h)}{h}=\vec 0.$$
If $f'(a)$ exists and is non-zero, is it possible $N$ to be non-linear? If so, please provide examples of $f$ and $N$, and specify $a$.
Aside: It's possible for $N$ to be linear, in which case $N$ is unique and is the differential $N(h)=hf'(a)$.
Of course it is: you can just add $g(h) h$ to $f'(a) h$, where $g(h) \to 0$ as $h \to 0$. An obvious example is $ \lvert h \rvert^{1+\alpha} b $, where $\alpha>0$ and $b$ is some constant of the right sort.