How many sizes of infinity are there?
In set theory, the different sizes of infinity are called cardinals. There are infinitely many cardinals.
What is a totally ordered set?
A totally ordered set is a set where the elements of that set are ordered so that for each pair of elements x,y either x<y or x=y or x>y. For example, the real numbers form a totally ordered set.
Does there exist a continuous function which is not differentiable?
Yes. The absolute value function is continuous everywhere, but the absolute value function is not differentiable at (0,0) since the graph of the absolute value function has a sharp change in direction at (0,0).