Let $K \subset BV([0,1]; \mathbb{R}^d)$ be a compact set of continuous bounded-variation paths (in a topology under which the signature map is continuous, e.g. the $1$-variation topology), and assume $K$ is chosen so that distinct paths in $K$ have distinct signatures (for instance, $K$ consists of paths with a fixed monotone time component, ruling out tree-like equivalences). Let $F: K \to \mathbb{R}$ be a continuous function. Then for every $\varepsilon > 0$ there exist a truncation level $N \geq 0$ and a linear functional $\ell$ on the truncated tensor algebra $T^N(\mathbb{R}^d) = \bigoplus_{k=0}^N (\mathbb{R}^d)^{\otimes k}$ such that \begin{align*} \sup_{x \in K} \left| F(x) - \ell(\mathrm{Sig}^{\leq N}(x)) \right| < \varepsilon, \end{align*} where $\mathrm{Sig}^{\leq N}(x)$ denotes the projection of $\mathrm{Sig}(x)$ onto $T^N(\mathbb{R}^d)$.