Given a sequence $t_n \rightarrow 0$ of pairwise distinct positive real numbers and a sequence $x_n \rightarrow x$ in a Hausdorff locally convex space $X$, is there a continuously differentiable function $g:\mathbb R \to X$ such that $g(t_n)=t_nx_n$ and $g'(0)=x$?
Motivation
As it is shown in [Averbukh V.I., Smolyanov O.G. (1968). The various definitions of the derivative in linear topological spaces // Russian Math. Surveys 23:4. P.67–113], the following two definitions of Hadamard differentiability of a function $f:X\to \mathbb R$ at $0$ are equivalent.
(1) There is a continuous linear functional $A \in L(X,\mathbb R)$ such that for any convergent sequence $t_n \rightarrow 0$ of positive real numbers and a convergent sequence $x_n \rightarrow x$ in $X$, $\lim_{n \to +\infty} \frac {f(t_nx_n)-f(0)} {t_n} - A(x_n)=0$.
(2) There is a continuous linear functional $A \in L(X,\mathbb R)$ such that for any differentiable function $g:\mathbb R \to X$ with $g(0)=0$ the function $f \circ g$ is differentiable at $0$ and $(f \circ g)'(0)=A(g'(0))$.
Thus, I wonder can the condition of differentiability of $g$ in (2) be replaced with continuous differentiablility without changing the result?