John Lee's Introduction to Topological Manifolds gives the following definition for a simplicial map:
"Suppose $K$ and $L$ are simplicial complexes [in $\mathbb{R}^n$]...A simplicial map from $K$ to $L$ is a continuous map $f : |K| \to |L|$ whose restriction to each simplex $\sigma \in K$ agrees with an affine map taking $\sigma$ onto some simplex in $L$."
Is it necessary to assume continuity, though? I think it follows from the other property: The simplices of $K$ form a locally finite closed cover of $|K|$, and since the restriction of $f$ to each simplex is continuous, we should be able to conclude that $f$ is continuous. Is this right?