Real Analysis/Limits and Continuity Exercises/Hints

\mathbb{R}$$ and next show that every continuous function is determined by its values on $$\mathbb{Q}$$
 * 1) No hint.
 * 2) You may want to prove first that the region above a convex function is convex (i.e. any straight line joining two points in the region, lies wholly in the region) and then using this fact argue by way of contradiction to show that convex functions are indeed continuous (i.e. no jump or removable discontinuity)
 * 3) Consider the function $$h(x) = f(x) - x$$. Using the Intermediate Value Property, show that $$\exists p$$ such that $$h(p) = 0$$.
 * 4) First show that the set of all infinite sequences of real numbers has the same cardinality as $$
 * 1) No hint.
 * 2) (a) Use mean value theorem, once we cover it. (b) Let $$\delta = \epsilon / K$$.