  Garret Birkhoff's HSP theorem characterizes the classes of models of
algebraic theories as those being closed with respect to homomorphic images,
subalgebras, and products. In particular, it implies that an algebra
$\mathbf{B}$ satisfies all equations that hold in an algebra $\mathbf{A}$ of
the same type if and only if $\mathbf{B}$ is a homomorphic image of a
subalgebra of a (possibly infinite) direct power of $\mathbf{A}$. The former
statement is equivalent to the existence of a natural map sending term
functions of the algebra $\mathbf{A}$ to those of $\mathbf{B}$, and it is
natural to wonder about continuity properties of this mapping. We show that
this map is uniformly continuous if and only if every finitely generated
subalgebra of $\mathbf{B}$ is a homomorphic image of a subalgebra of a finite
power of $\mathbf{A}$ -- without any additional assumptions concerning the
algebras $\mathbf{A}$ and $\mathbf{B}$. Moreover, provided that $\mathbf{A}$ is
almost locally finite (for instance if $\mathbf{A}$ is locally oligomorphic or
locally finite), the considered map is uniformly continuous if and only if it
is Cauchy-continuous. In particular, our results extend a recent theorem by
Bodirsky and Pinsker beyond the countable $\omega$-categorical setting.
