Many control problems are naturally expressed in continuous time. Yet, in Iterative Learning Control of linear systems, sampling the output signal has proven to be a convenient strategy to simplify the learning process while sacrificing only marginally the overall performance. In this context, the control action is similarly discretized through zero-order hold - thus leading to a discrete-time system. With this paper, we want to investigate an alternative strategy, which is to track sampled outputs without masking the continuous nature of the input. Instead, we look at the whole input evolution as an element of a functional subspace. We show how standard results in linear Iterative Learning Control naturally extend to this context. As a result, we can leverage the infinite-dimensional nature of functional spaces to achieve exact tracking of strongly non-square systems (number of inputs less than outputs). We also show that constraints - like those imposed by intermittent control - can be naturally integrated within this framework.