text:
\section{The groups for the immersion}

We present some of the notation that will be used throughout this article.

We fix a partition $\{P_0, P_1\}$ of $\mathfrak c$ such that $|P_0|=|P_1|=\mathfrak c$, and $\omega+1\subseteq P_1$.
Let $\{R_0,R_1\}$ be a partition of $2^\mathfrak c \setminus \mathfrak c$ such that $|R_0|=|R_1|=2^\mathfrak c$. Define
$\mathbb U= \mathbb{Q}^{(R_0)}\oplus\mathbb{Q}^{(R_1)}$ and $\mathbf{U}= (\mathbb{Q}/\mathbb{Z})^{(R_0)}\oplus\mathbb{Q}^{(R_1)}$.

We define $\mathbb W = (\mathbb{Q} /\mathbb{Z})^{( P_0 \times \omega)} \oplus \mathbb{Q}^{ (P_1)} \oplus \mathbf{U}$. We also let $\mathbb X= \mathbb{Q} ^{( P_0\times \omega)} \oplus \mathbb{Q}^{ (P_1)} \oplus \mathbb{U}$.

Throughout this work, the main group will be $\mathbf X = \oplus_{n >1,m \geq 1 } (\mathbb Q/\mathbb Z)^{(C_{n,m} \times m)}\oplus \mathbb Q^{(P_1)}\oplus \mathbf{U}$, where $\{ C_{n,m}:\, n>1, m\geq 1\}$ is a partition of $P_0$ into pieces of cardinality $\mathfrak c$ and $C_0=\bigcup_{m,n>1}C_{n,m}$ is such that $P_0\setminus C_0=\bigcup_{n>1}C_{n,1}$ has cardinality $\mathfrak c$. We will also define  $C_1 \subseteq P_1$ with $|C_1|=\mathfrak c$, $\omega\subseteq C_1$ and $|P_1 \setminus C_1|=\mathfrak c$.
To simplify the notation of the forcing later we will also
partition $C_1$ as $\{C_{1,m}:\, m>1\}$ and let $C_{1,1}=P_1 \setminus C_1$.

\subsection{Structure of the article}
 We use forcing to construct an injective group homomorphism $\Phi :\, {\mathbf X}\to \mathbb{T}^{\mathfrak c}$. The range of this homomorphism will be countably compact and have convergent sequences. Each forcing condition will be a partial countable piece of this homomorphism, and the existence of suitable conditions was proved in \cite{bellini&boero&castro&rodrigues&tomita}.

Of course, not every subgroup of $X$ is countably compact. However, we show that if $H$ is a group such that $2^\mathfrak c \geq |H| = r(H) = \mathfrak{c}$ and for all $d, n \in \mathbb{N}$ with $d \mid n$, the group $dH[n]$ is either finite or has cardinality at least $\mathfrak{c}$, then it is isomorphic to a subgroup of $\mathbf X$ that is countably compact and has convergent sequences considering the subspace topology of $X$ generated by $\Phi$. To show that such a copy exists, we define the concept of \textit{large subgroup of } $X$, which was inspired by the concept of \textit{nice immersion} we defined in \cite{bellini&boero&castro&rodrigues&tomita}. This concludes the classification.

The convergent sequences will be the sequences $(\frac{n!}{S} \chi_n:\, n \in \omega )$ in $G\subseteq {\mathbf X}$ (identifying $G$ with its copy in ${\mathbf X}$), for each positive integer $S$ and they will converge to $0$.

\subsection{More notation}

Given $w \in \mathbb{W}$ or $w \in \mathbb X$, $x \in (P_0 \times \omega) \cup R_0$ and $y \in  P_1 \cup R_1$, we denote by $w(x)$ the $x$-th coordinate of $w$ and $w(y)$ the $y$-th of $w$, so the functions $w\rightarrow w(x)$ and $w\rightarrow w(y)$ are the natural projections.

We also fix well defined numerators and denominators for fractions: if $r \in  \mathbb Q/\mathbb Z$, then $p(r)$ and $q(r)$ are the unique integers $p, q$ such that $q>0$, $\gcd(p, q)=1$, $0\leq p<q$ and $r=\frac{p}{q}+\mathbb Z$. Likewise, if $r \in \mathbb Q$,  $p(r)$ and $q(r)$ are the unique integers $p, q$ such that $q>0$, $\gcd(p, q)=1$ and $r=\frac{p}{q}$.

Given $w \in \mathbb{W}$ (or $w \in \mathbb X$), we denote by $w^0$ and $w^1$ the unique elements of $\mathbb{W}$ (or $\mathbb X$) such that $\supp w^0 \subseteq ((P_0 \times \omega) \cup R_0)$, $\supp w^1 \subseteq P_1 \cup R_1$ and $w=w^0+w^1$, that is, $w\rightarrow w^0$ is the natural projections into $(\mathbb Q/\mathbb Z)^{((P_0 \times \omega) \cup R_0)}$ (or $\mathbb Q^{((P_0 \times \omega )\cup R_0)}$) and $w \rightarrow w^1$ is the natural projection into $\mathbb Q^{(P_1\cup R_1)}$. Also, we call $w^{1,0}$ and $w^{1,1}$ the natural projections of $w$ onto $\mathbb Q^{(\omega)}$ and $\mathbb Q^{((P_1 \cup R_1)\setminus \omega)}$, respectively.

We also define $p(w) = \max\{|p(w(z))| : z \in \supp w\}$  and $q(w) = \max\{q(w(z)) : z \in \supp w \}$ if $w \neq 0$. We define $p(0)=0$ and $q(0)=0$.

Similarly, given $g:\omega\to \mathbb{W} $ (or $\mathbb X$), we define $g^0$, $g^{1}$, $g^{1, 0}$ and $g^{1, 1}$. So $g=g^0+g^1=g^0+g^{1, 0}+g^{1, 1}$, $\supp g^0\subseteq ((P_0\times \omega) \cup R_0)$, $\supp g^1\subseteq P_1 \cup R_1$, $\supp g^{1, 0}\subseteq \omega$ and $\supp g^{1, 1}\subseteq (P_1\cup R_1)\setminus \omega$; where  $\supp h = \bigcup\{ \supp h(k):\, k\in \omega\}$  for a sequence $h$.

It will be useful to be able to easily transform an element of $\mathbb X$ into an element of $\mathbb{W} $. Thus, given $w \in \mathbb X$, we define $[w]$ as the unique element of $\mathbb W$ such that for every $x \in (P_0\times \omega) \cup (R_0)$, $[w](x)=w(x)+\mathbb Z$ and for every $y \in P_1 \cup R_1$, $[w](y)=w(y)$. Clearly, the function $w\rightarrow [w]$ is a group homomorphism from $\mathbb X$ onto $\mathbb{W}$. Given a function $g:\omega\rightarrow \mathbb X$, we also define $[g]:\omega\rightarrow \mathbb{W}$ be given by $[g](n)=[g(n)]$ for every $n \in \omega$.


TITLE: Existence of a Non-Linear Function Satisfying Certain Conditions
QUESTION [1 upvotes]: 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 $N(h)=hf'(a)$.

REPLY [1 votes]: 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.