I am reading Arnold's ODE (3rd edition) and I am confused with the remark.
Page 62 :
Remark. The condition of smooth dependence on the time t is needed in order to eliminate pathological examples such as the following:
let { $\alpha$ } be a basis of the group R i.e., a set of real numbers such that each real number has a unique representation in the form of a finite linear combination of numbers of the set with integer coefficients.
Q1.) I think he is not talking about the vector basis since the integer is not even a field.
How can I show there indeed exists such "basis" ?
To each number $\alpha$ of the basis we assign the translation of the line by some distance, paying no attention to other elements of the basis. Setting $g^{\alpha_1+...+\alpha_k}=g^{\alpha_1}...g^{\alpha_k}$, we obtain a one-parameter transformation group each of whose element is a translation of the line consequently a differmorphism; but in general $g^t$ is not a smooth function of t and is even discontinuous.
Q2.) I don't know how this reasoning shows $g^t$ must be smooth with respect to t, as he remarks at the beginning.
Instead of smoothness with respect to t one may require only continuous(from which smoothness is a consequence) but we have no need to do this.
Q3.) I don't understand this either. Please give me a hint.
Regarding Q1), I think that you cannot show anything, because such a basis does not exist. Indeed, suppose it exists, and pick some (necessarily nonzero) $\alpha$ belonging to it. $\alpha/2$ is a nonzero real number, so it must be uniquely expressed in the form $$ \frac{\alpha}{2} = k_1 \alpha_1 + \ldots + k_n \alpha_n, $$ where $k_1, \dots, k_n \in \mathbb{Z} \setminus \{0\}$, $\alpha_1, \dots, \alpha_n$ belong to the basis. Then we have $$ \alpha = (2k_1) \alpha_1 + \ldots + (2k_n) \alpha_n, $$ which means that $\alpha$ has two representations, a contradiction.
I would look for a such a pathological one-parameter group in the following way. Take any non-measurable solution of Cauchy's functional equation $h(t + s) = h(t) + h(s)$, and define $$ g^t(x) = e^{h(t)}x, \quad x \in \mathbb{R}. $$
EDIT: We have obtained a homomorphism of the additive group of reals into the groups of scalings of $\mathbb{R}$. I chose it since it looks to me more natural in the context of linear differential equations. However, to remain in the setup of the textbook one should rather choose a homeomorphism of the additive group of reals into the groups of translations of $\mathbb{R}$. Then $$ g^t(x) = x + h(t), \quad x \in \mathbb{R}. $$
Now, you can find answers to Q2) and Q3) in Overview of basic facts about Cauchy functional equation.