For each description below, provide a specific example fitting thedescription (p

Question

For each description below, provide a specific example fitting thedescription (provide some
justification), or else explain why no such example exists.
Sets
1. A proper open set which is not an interval
2. A countably infinite closed set
3. A countable open set
4. A infinite collection of open sets whose union is [-3; 8]
5. A infinite collection of closed sets whose intersection is [-3;8]
6. An open set S with a point (in S) that is isolated from S.
7. A set with exactly 4 accumulation points
8. A compact, countable, disconnected set
Functions
9. A function which is continuous everywhere, except at sqrt2
10. A function which is differentiable everywhere, except at sqrt2,and which is continuous everywhere
11. A continuous function f such that the inverse image of f,f<- [[2; 3]] = [-4;-3] U [3; 4]
12. A continuous function f on R and an open set V of R such thatf(V ) is not open

13. Let f : X --> Y be any function (not necessarilycontinuous). Show carefully that
f [S^c] = (f [S])^c for any subset S belonging to Y . (S^c = Y/Shere). (^c represents the complement)

14. Let S = {1} U {(n+1)/n | n belongs to the natural numbers N};and let f be a continuous function whose domain contains S. Showthat f attains a maximum on S.

15. (Adapted from the NPR radio show Car Talk"). A Tibetan monkleaves one morning
at 6 AM from the valley for the monastery in the mountains, whichshe reaches at 6 PM.
The next day, she leaves the monastery at 6 AM and reaches thevalley at 6 PM, using the
same path. Show that there is a place on the path that the monkreaches at the same time
on both days. (You will need to assume that the monk has notreached enlightenment yet
and thus cannot levitate. You might also want to check Presentation17).

Presentation 17
a. Show that, if f is continuous, then the image of a connected setby f is connected.
b. Show that, if f is continuous, then the image of a compact setby f is compact.
c. Show that if f is continuous on interval [a; b], then f achievesa maximum and a
minimum on [a; b], achieves every value between its maximum andminimum on [a; b]. (The first statement is called the Extreme valueTheorem, the second the Intermediate Value Theorem)

16. Let f and g be two continuous functions on R. Show that f- g is continuous.

Explanation / Answer

you should prolly repost this as at least 3 or 4 posts cause thisinvolves a lot of work!

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