However, apparently they are not continuous . The Real Numbers are all the numbers on the number line. You will explore this concept further in Algebra II and Calculus. Suffice it to say now that if you can plot the function without picking up your pencil, it is continuous. A number line is such a plot.
Real Numbers are either rational or irrational.
All rational and all irrational numbers are real numbers. The rational and irrational numbers are disjoints sets which together make up the real numbers. Then there exists a pairing of each number such that neither set has any elements left over. But we will now show that there is at least one real number which is not included in this pairing. Thus the set of real numbers is non-denumerable.