Let $c(t)$ be an everywhere positive real function of $t$.
Let also $F(t)$ and $G(t)$ be vectors of some (fixed) topological vector space. Let the functions $F$ and $G$ be continuous and $F(t)$, $G(t)$ be nonzero everywhere.
From the equality $G(t)=c(t)F(t)$ (for every $t$) can we deduce that $c$ is a continuous function?
If it does not hold for arbitrary topological vector space, does it hold for special cases (e.g. for $\mathbb{R}^n$)?
Since $$ c(t) = \frac{\| F(t) \|}{ \|G(t) \|} $$ and each of the functions on the right ($F, G, \|\cdot \|$} is continuous, as is the quotient $Q(x, y) = x/y$, at least away from $y = 0$, the function $c$ is continuous (at least for $\mathbb R^n$, where norm is defined).
As for arbitrary topological vector spaces --- I suspect that it might be true there as well, but have no real insight; merely (slightly) educated guesswork.