I don't think this is right at all. Digital computers express non-continuous functions, and they sometimes use those to approximate continuous functions.

For example, for a function f(x) defined on R with f(x) = -x if x < 0 and f(x) = 7+x if x >= 0, how would you approximate it by a continuous function g(x) with precision lower than, say, 1 (i.e. |f(x) - g(x)| < 1 for any x in R)?

And of course, there are functions with much worse discontinuities than this.