From time to time, I have come across course ideas emphasizing the discrete over the continuous, such as Peter Saveliev's Fantasy Math curriculum (update: see also his draft book on discrete calculus) and Kelemen et al's article Has Our Curriculum Become Math-Phobic?. For instance, Saveliev refers to discrete functions and data coming in digital format (no formula, just function), leading to a discrete calculus based on discrete differential forms. Kelemen et al discuss early discrete mathematics for computer science students.
Questions: What are the pros and cons of promoting the discrete and, to some extent, de-emphasizing or postponing the continuous in the first year or two of undergraduate education? What are your experiences, if any, of teaching in such an environment?
See also Why does undergraduate discrete math require calculus? on Math Overflow Stack Exchange back in 2010.
Speculatively, it might be possible, albeit unconventional, to teach an early undergraduate course using graph theory as a springboard to introduce a range of concepts, such as linear algebra and network analysis (e.g. Kirchhoff's circuit laws), algorithms, mathematical induction, simplicial complexes, elementary combinatorial topology, relevant aspects of discrete/computational geometry, and possibly some elements of discrete calculus. The topics could be kept as low-dimensional, visual and concrete as feasible. Graph theory is fairly rich in applications these days, so the course could potentially appeal to those with both pure and applied interests.