  $\text{TT}^{\Box}_{{\mathcal C}}$ is a generic family of effectful,
extensional type theories with a forcing interpretation parameterized by
modalities. This paper identifies a subclass of $\text{TT}^{\Box}_{{\mathcal
C}}$ theories that internally realizes continuity principles through stateful
computations, such as reference cells. The principle of continuity is a seminal
property that holds for a number of intuitionistic theories such as System T.
Roughly speaking, it states that functions on real numbers only need
approximations of these numbers to compute. Generally, continuity principles
have been justified using semantical arguments, but it is known that the
modulus of continuity of functions can be computed using effectful computations
such as exceptions or reference cells. In this paper, the modulus of continuity
of the functionals on the Baire space is directly computed using the stateful
computations enabled internally in the theory.
