Before revision:  As a first contribution, it presents a proof confirming the conjecture in Pearl (1995), showing that \citet*{pearl1995testability}  the validity of an instrument indeed cannot be tested in the most general case when the endogenous variable is continuous. However, as a second contribution, it shows that already weak restrictions on the instrumental variable modelreestablish theoretical testability . Continuity is already enough. Monotonicity introduces further testable implications .
After revision:  As a first contribution, it presents a proof confirming the conjecture in Pearl (1995), showing that \citet*{pearl1995testability} the validity of an instrument indeed cannot be tested in the case where the endogenous treatment is continuously distributed. This stands in contrast to the classical testability results for instrument validity when the treatment is discrete. However, imposing weak structural assumptions on the model, such as continuity between the observable variables, can re-establish theoretical testability in the continuous setting .