- #1
Checkfate
- 149
- 0
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 [tex] f(x)=1-\frac{1}{x-4} [/tex], 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 [tex]\lim_{x\rightarrow a}f(x)=f(a)[/tex]... Which was stated in my text as meaning that the function is continuous... which it obviously isnt.
[tex]\lim_{x\rightarrow a}f(x)=\lim_{x\rightarrow a}(1-\frac{1}{x-4})[/tex]
[tex]=1-\lim_{x\rightarrow a}\frac{1}{x-4}[/tex]
[tex]=1-\frac{1}{a-4}[/tex]
[tex]=f(a)[/tex]
Can someone cure my confusion? Thanks guys.
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 [tex] f(x)=1-\frac{1}{x-4} [/tex], 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 [tex]\lim_{x\rightarrow a}f(x)=f(a)[/tex]... Which was stated in my text as meaning that the function is continuous... which it obviously isnt.
[tex]\lim_{x\rightarrow a}f(x)=\lim_{x\rightarrow a}(1-\frac{1}{x-4})[/tex]
[tex]=1-\lim_{x\rightarrow a}\frac{1}{x-4}[/tex]
[tex]=1-\frac{1}{a-4}[/tex]
[tex]=f(a)[/tex]
Can someone cure my confusion? Thanks guys.
Last edited by a moderator: