Continuous formulations are used with digital computers all the time. Limited precision of floats sometimes causes numerical instability for some algorithms, but usually these are fixable with different (sometimes less efficient) implementations.

Discretizing e.g. time or space is perhaps a bigger issue, but the issues are usually well understood and mitigated by e.g. advanced numerical integration schemes, discrete-continuous formulations or just cranking up the discretization resolution.

Analytical tools for discrete formulations are usually a lot less developed and don't as easily admit closed-form solutions.