How does a computer interpret real numbers?

Question

I understand that the modern day digital computer works on the binary number system. I can also get, that the binary representation can be converted to rational numbers.

But I want to know how does the present day computational model interpret real numbers.

For eg:

On a daily basis we can see that a computer can plot graphs. But here, graphs may be continuous entities. What is the mathematical basis, that transforms a discrete (or countable, at most) like the binary system to something mathematically continuous like a say, the graph of $f(x) = x$.