
Post by Peiley on Jul 21, 2015 19:48:41 GMT
Could you clarify what sort of counterexample 3(c) is asking for?
It seems that in part 3(a) and 3(b), both of the sets are disconnected. So as a counterexample, is 3(c) asking for an example where the subset intersects with X, but yet is still everywhere positive but discontinuous at one point?



Post by KBeal on Jul 22, 2015 3:14:02 GMT
Let me know if this helps:
Let S:=H\cup X. a) if H= graph(f) for f:(0,\infty)\to\mathbb{R}, f(x):=1/x Is S connected? b) if H= graph(f) for f:\mathbb{R}\to (0,\infty), f continuous Is S connected? c) if H= graph(f) for f:\mathbb{R}\to (0,\infty), \exists x_0\in\mathbb{R} such that f is continuous on \mathbb{R}\backslash\{x_0\}, is S connected? No. Let f(x):=blahblahblah. Then f is continuous on \mathbb{R} except at x_0=blah, but S is not connected.
(so a and b are true/false whereas c tells you that the statement is false)



Post by Guest on Jul 22, 2015 3:20:46 GMT
Hey Khalilah,
So are you saying a) and b) are true or false questions whereas c) is telling you that S is definitely disconnected in the case where f is everywhere positive but discontinuous at just one point, just give an example of such f?
Thanks



Post by KBeal on Jul 22, 2015 15:27:09 GMT
Yes. To repeat/clarify, the problem has 3 parts (a, b, and c). The problem is based on the following thms: If f:M to N is continuous and M is connected, then graph(f) is connected. The union of two connected sets with nonempty intersection is connected.
(We did a generalization of this thm in class. We had a connected set K and a collection of connected sets S_\alpha and each S_\alpha had at least one point in common with K. The thm said that in this case, K unioned with all the S_\alpha is a connected set. )
The problem is investigating these thm because the problem investigates the following: you have a continuous function (or continuous except at one point) on a connected domain, and you want to see if graph(f) union xaxis is connected.
part a gives you a specific function: f is continuous on its domain, f is positive (so graph(f) never touches xaxis), and domain of f is connected. IN THIS CASE, the union of graph f with the xaxis is connected (even though the intersection of graph f and xaxis is the emptyset) [proof by you]
part b gives you a class of functions: any function f s.t. f is continuous on its domain, f is positive, and domain of f is (\infty, \infty). Is the union of graph (f) and xaxis connected?
part c gives you a class of functions: any fnc f such that f is continuous, except at 1 point, f is positive, and domain of f is (\infty,\infty). (So, for instance, f could be defined by f(x)=1/x if x is not 0, and f(0)=1.) Now part c is telling you that if f is continuous on (\infty,x_0)and(x_0,\infty) and f is positive, then graph(f)+xaxis may or may not be connected. (in fact, graph(f)+xaxis will ALWAYS be disconnected. You are not asked to prove this, though.) You are asked to give an example of a function f such that 1. f is continuous, except at 1 precisely one point (maybe the origin? may as well assume the discontinuity of the function occurs at the origin. this is because if the discontinuity occurs at x_0,then translate it left by x_0 units and you get a function with its discontinuity at the origin. so, without loss of generality, the discontinuity occurs at the origin.)
2. f is positive 3. the union of graph(f) and the xaxis is not connected



Post by KBeal on Jul 22, 2015 16:29:14 GMT
(in fact, graph(f)+xaxis will ALWAYS be disconnected. You are not asked to prove this, though.)
Here is an example: f(x)=  1/(1x) if x is less than or equal to 0
 2+sin{1/x} otherwise
 2/[x+1(1/pi)] if x is greater than or equal to 1/pi
In this example, f is continuous except at x=0, f>0, and the union of xaxis and graph(f) is connected.



Post by KBeal on Jul 23, 2015 1:07:56 GMT
Please see announcements for correction to earlier posts

