  In the setting of constructive pointfree topology, we introduce a notion of
continuous operation between pointfree topologies and the corresponding
principle of pointfree continuity. An operation between points of pointfree
topologies is continuous if it is induced by a relation between the bases of
the topologies; this gives a rigorous condition for Brouwer's continuity
principle to hold. The principle of pointfree continuity for pointfree
topologies $\mathcal{S}$ and $\mathcal{T}$ says that any relation which induces
a continuous operation between points is a morphism from $\mathcal{S}$ to
$\mathcal{T}$. The principle holds under the assumption of bi-spatiality of
$\mathcal{S}$. When $\mathcal{S}$ is the formal Baire space or the formal unit
interval and $\mathcal{T}$ is the formal topology of natural numbers, the
principle is equivalent to spatiality of the formal Baire space and formal unit
interval, respectively. Some of the well-known connections between spatiality,
bar induction, and compactness of the unit interval are recast in terms of our
principle of continuity.
  We adopt the Minimalist Foundation as our constructive foundation, and
positive topology as the notion of pointfree topology. This allows us to
distinguish ideal objects from constructive ones, and in particular, to
interpret choice sequences as points of the formal Baire space.
