Equality of limit in path connected space

Question

I have a topological vector space $(V, \mathcal T)$ such that there exists a smooth path between any two elements of $V$. I want to compute the limit of a function $F:V\rightarrow W$. Is it true that $$\lim_{x\rightarrow v} F(x) = \lim_{t\rightarrow 0}F(\varphi(t)),$$ where $\varphi$ is a smooth path such that $\varphi(0) = v$ and $\varphi(1) = x$? In other words, instead of going from $x$ to $v$ I follow the path $\varphi$? Intuitively, this seems correct but I could not prove it (except using an "informal" argument like $\lim_{x\rightarrow v}F(x) = \lim_{\varphi(1)\rightarrow\varphi(0)}F(x)$ and $\varphi(1)\rightarrow \varphi(0)$ "is the same as $t\rightarrow 0$").

Answer 1

The answer is in general no: Let $\varphi:[0,1]\to V$ a smooth path such that $\varphi(0)=v$. Let $\eta:[0,1]\to [0,1]$ a smooth function such that $\eta(t)=0$ for $0\leq t<1/2$ and $\eta(1)=1$, then $\psi=\varphi\circ\eta:[0,1]\to V$ is a smooth path such that $\psi(t)=v$ for every $t\in [0,1/2[$. Thus for every $0<t<1/2$ we have that $$ F(\psi(t)) = F(v) $$ and thus $$ \lim_{to\to 0^+} F(\psi(t)) = F(v) $$ which is distinct from $\lim_{x\to v}F(x)$ if $F$ is not continuous at $v$.

But we can add the following hypothesis: There is $\rho>0$ such that for $0<t<\rho$, $\varphi(t)\neq \varphi(0)=v$. Also, assume that a limit $\lim_{x\to v}F(x)$ exists. In this case let $L$ be a limit of $F$ as $x\to v$.

Let $\varphi:[0,1]\to V$ be a smooth path such that $\varphi(0)=v$. Consider the composition $F\circ\varphi:[0,1]\to W$. Let $V'$ be a neighborhood of $L$ in $W$. By the definition of limit, there is a neighborhood $U$ of $v$ such that $$ F(U\setminus\{v\})\subseteq V'. $$ Now, as $\varphi$ is smooth, it is continuous, and as $U$ is a neighborhood of $\varphi(0)=v$ there is a $\delta_1>0$ such that $$ \varphi(t)\in U \quad \text{ for all } 0\leq t<\delta_1. $$ Let $\delta=\min\{\delta_1,\rho\}$ and let $0<t<\delta$, then $\varphi(t)\neq v$ by the additional hypothesis and $\varphi(t)\in U$, thus $$ \varphi(t)\in U\setminus\{v\}. $$ As $F(U\setminus\{v\})\subseteq V'$, we have that $$ F(\varphi(t))\in V' \quad \text{for all } t \text{ such that } 0<t<\delta $$ which means that $L$ is a limit of $F(\varphi(t))$ as $t\to 0^+$.

Also, note that we only need the continuity of $\varphi$.