Proving Function Continuity in [-1,30]: Understanding the Example

[Checkfate]

I am trying to fully understand this example from a textbook I am reading:

http://img59.imageshack.us/img59/9237/continuityyn8.jpg

What I am not understanding is how they are proving it for [-1,1].. The way I see it is they proved that the function is continuous for all values in it's domain...

For example, I thought up this problem on my own to help me understand :

Given $$f(x)=1-\frac{1}{x-4}$$, prove that f(x) is continuous in the interval [-1,30] (Obviously it's not continuous at x=4.) The problem is that I don't see the connection between the interval and the solution...

I can just go ahead and prove that $$\lim_{x\rightarrow a}f(x)=f(a)$$... Which was stated in my text as meaning that the function is continuous... which it obviously isnt.

$$\lim_{x\rightarrow a}f(x)=\lim_{x\rightarrow a}(1-\frac{1}{x-4})$$
$$=1-\lim_{x\rightarrow a}\frac{1}{x-4}$$
$$=1-\frac{1}{a-4}$$
$$=f(a)$$

Can someone cure my confusion? Thanks guys.

 

[Checkfate]

I am not sure as to whether or not the latex is really screwed up in my post, I edit it and it looks completely diff from what I see when I refresh the post. Click on the indivual latex box's to see what I meant to type if it comes up screwy. Maybe it's just my pc.

Nevermind, it's fine now.

 

[Checkfate]

oh my.. I see it now, limf(x)=f(a) for all values except a=4... since that would result in an undefined statement. But I still don't understand how they are testing for the interval [-1,1]... How would I test my made up example in the interval [-1,1]?

 

[arildno]

They are, indeed, showing it for the values in the function's PRESCRIBED DOMAIN, that is the interval from -1 to 1

 

[StatusX]

The only time they implicity use the assmption that x was in [-1,1] was in moving the limit inside the square root. sqrt(x) is only continuous for x>=0, (namely because it is only defined here), so they needed that fact that 1-x^2 was non-negative.

 

[Checkfate]

Okay I understand what they were doing now, thanks :)

edit:Nevermind this other question, I reread the definition/example and it became clear.