
real analysis
hello. just want to ask some help with these, or a sketch of the proof perhaps:
1. how can i prove the bolzano intermediate value theorem?
2. if K is compact and f is continuous on K, then f(K) is compact. is there another way of proving this without using the heineborel theorem? but it would be ok to use that theorem to.
3. what could be another criterion for a product of uniformly continuous functions f and g on X to uniformly continuous on X too?
4. i would like to verify if this proof for the continuity of thomae's function is ok, or do i have to add or cancel something:
Proof of Thomae's Function:
Let a be an element of {Q intersection [1,0]}, there exists {x_{n}} subset of {Q' intersection [0,1]} such that x_{n} approaches a (by density theorem). Lim f(x_{n}) = 0 but f(a) = 1/n. Hence, f(x) is not continuous at rational points.
On the other hand, let a' be an element of {Q' intersection [1,0]}, there exists {x_{n}} subset of {Q intersection [0,1]} such that x_{n} approaches a' (by density theorem). Lim f(x_{n}) = 1/n approaching 0 and f(a') = 0. Hence, f(x) is continuous at irrational points.