In this work, we are concerned with the concepts ofF -α-convergence,F -pointwise convergence and F -uniform convergence for sequences of functions on metric spaces, where F is a filter on N. We define the concepts of F -limit function, F -cluster function and limit function respectively for each of these three types of convergence, and obtain some results about the sets of F -cluster and F -limit functions for sequences of functions on metric spaces. We use the concept of F -exhaustiveness to characterize the relations between these points