1. ## Continuity problem

I'm not sure how to find these intervals of continuity:

ln(x-2)

sin(x^2 - 2)

I don't understand how I am supposed to find the continuity. I look at the graphs of each and I'm not sure what to look for. I understand functions like tan(x), etc. but with these I'm a little confused. Can anyone point me in the right direction?

2. Originally Posted by nautica17
I'm not sure how to find these intervals of continuity:

ln(x-2)

sin(x^2 - 2)

I don't understand how I am supposed to find the continuity. I look at the graphs of each and I'm not sure what to look for. I understand functions like tan(x), etc. but with these I'm a little confused. Can anyone point me in the right direction?
these functions are continuous where they are defined. the log function is continuous where its argument is positive: for example, $\ln x$ is continuous where $x > 0$

the sine function is continuous everywhere, and so it is continuous wherever its argument is: $\sin (f(x))$ is continuous where $f(x)$ is continuous.

so what can you do with that information?

3. Originally Posted by Jhevon
these functions are continuous where they are defined. the log function is continuous where its argument is positive: for example, $\ln x$ is continuous where $x > 0$

the sine function is continuous everywhere, and so it is continuous wherever its argument is: $\sin (f(x))$ is continuous where $f(x)$ is continuous.

so what can you do with that information?
Okay, so from what you have said I have come up with this:

ln(x-2) is continuous when x is greater than 2?

sin(x^2 - 2) is continuous no matter what and on whatever interval you sort of choose to put it on?

4. Originally Posted by nautica17
Okay, so from what you have said I have come up with this:

ln(x-2) is continuous when x is greater than 2?

sin(x^2 - 2) is continuous no matter what and on whatever interval you sort of choose to put it on?
Correct.

5. Alright thanks guys.

Sorry if it's a simple question, but it looked confusing at first.

6. Originally Posted by Jhevon
these functions are continuous where they are defined. the log function is continuous where its argument is positive: for example, $\ln x$ is continuous where $x > 0$

the sine function is continuous everywhere, and so it is continuous wherever its argument is: $\sin (f(x))$ is continuous where $f(x)$ is continuous.

so what can you do with that information?
hey isn't -1<=sin(x^2-2)<=1
=>-pi/2<=x^2-2<=pi/2
=>sqrt(2-pi/2)<=x<=sqrt(pi/2+2)????

7. Originally Posted by Pulock2009
hey isn't -1<=sin(x^2-2)<=1
=>-pi/2<=x^2-2<=pi/2
=>sqrt(2-pi/2)<=x<=sqrt(pi/2+2)????
What?!

8. Originally Posted by Pulock2009
hey isn't -1<=sin(x^2-2)<=1
=>-pi/2<=x^2-2<=pi/2
=>sqrt(2-pi/2)<=x<=sqrt(pi/2+2)????
No, that is an interval on which the function is one-to-one and has nothing to do with where it is continous. $x^2- 2$ is continuous for all x, sin(x) is continous for all x, and the composition of two continuous functions is continuous.

In a certain, very specific, sense, "almost all" functions are never continuous. But continuous functions are so nice that our ways of writing functions have developed so that functions written as a single "formula" are continuous where ever they are defined.