Combinations of Continuous Functions

(1) Let .

(a) Prove that is continuous at .

(b) Prove that is continuous at a point . (The identity ) will be helpful)

(2) Using the characterization of continuity (and thus using no previous results about sequences), show that the linear function is continuous at every point of .

(3)

(a) Using the definition: ( A function is continuous at a point if, for all , there exists a such that whenever (and ) it follows that . If is continuous at every point in the domain , then we say that is continuous on .) show that any function with domain will necessarily be continuous at every point in its domain.

(b) Show in general that if is an isolated point of , then is continuous at .

(4) Assume is continuous on and let . Show that is a closed set.

If anyone could figure out any of these, I would greatly appreciate it! Thanks!