PSet 5, #3c

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)

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 x-axis is connected.

part a gives you a specific function: f is continuous on its domain, f is positive (so graph(f) never touches x-axis), and domain of f is connected.
IN THIS CASE,  the union of graph f with the x-axis is connected (even though the intersection of  graph f and x-axis 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 x-axis 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)+x-axis may or may not be connected.
(in fact, graph(f)+x-axis 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 x-axis is not connected