Why are real numbers useful?

A question (by a fellow CS student taking a first course in calculus, presumably after the lecture in which continuity was introduced: was as follows.

In the real, physical world, we deal with numbers that are sort of “finite” or “discrete” by their nature; there's no such thing as a perfect circle in the physical world. In CS, we model computers with discrete mathematics and it’s enough. So why real analysis? Why concepts like continuity and “completeness” of real numbers are useful? Why do we need them?

I found Math.se has lots of questions for similar “concrete” justifications for complex numbers and great answers for them, but I didn’t really manage to find similar for this. The question Are all numbers real numbers? is related, but I’m not sure it’s exactly what I’m looking for.

My attempt at an answer was along the line:

Are we content with the length of the side of a square that has area of 2 units remaining a undefined number, even if we never manage find such a square?

and

Calculus and real analysis provide results that are useful even if in numerical calculations we use finite approximations. To really understand what's going on, we want it to be rigorous.

but I’m not sure if I was persuasive enough. Any better ideas?

It's far, far easier to understand some notion of "approximately a circle" if you are first able to understand some notion of "circle".

And even if you try to stick to, say, just the rational numbers, the real numbers keep managing to show up anyways. e.g. as soon as you ever find yourself in the situation where you think to identify a number by a function that tells you whether or not other rational numbers are larger or smaller than it, you've suddenly reinvented the real numbers.

It's telling that number theory -- one of the subjects that, a priori, should be one of the most discrete subjects of mathematics -- makes heavy use of the real numbers. And even that isn't enough 'continuous' mathematics for the sake of number theorists: they've invented $p$-adic numbers too!

Perhaps you could point your friend to the book Concrete Mathematics: A Foundation for Computer Science by Graham, Knuth & Patashnik, which shows a lot of useful connections between continuous mathematics (with real numbers) and discrete mathematics. ("Concrete = continuous + discrete".)

For example, the Euler–Maclaurin summation formula from advanced calculus can be used to derive approximations for discrete things like $\sum_{k=1}^n \frac1k$.