Calculus-I: discontinuity of piecewise function HW problem

[grey2q]

Screenshot of my homework problem along with my solution so far. I'm not sure if I'm doing this correctly and if I am... if I'm answering correctly. Thank you. (EDIT: I made 1 small error with the piecewise definition. Ignore the f(x) before g(x).)

 

[Mark44]

Screenshot of my homework problem along with my solution so far. I'm not sure if I'm doing this correctly and if I am... if I'm answering correctly. Thank you. (EDIT: I made 1 small error with the piecewise definition. Ignore the f(x) before g(x).)

View attachment 219851

What about where x < 0 or where x > 3? Your function is defined on those intervals, as well.

Also, the homework template you deleted is required here. In future posts, please don't delete this template.

 

[tnich]

Screenshot of my homework problem along with my solution so far. I'm not sure if I'm doing this correctly and if I am... if I'm answering correctly. Thank you. (EDIT: I made 1 small error with the piecewise definition. Ignore the f(x) before g(x).)

View attachment 219851

You have shown that the piecewise function is continuous at the boundaries of the pieces. If you are going to investigate the continuity of the function, you should also find out if there are any other discontinuities.

 

[grey2q]

What about where x < 0 or where x > 3? Your function is defined on those intervals, as well.

Also, the homework template you deleted is required here. In future posts, please don't delete this template.

Thanks for the response.

I think I understand. So, would it make more sense for me to say that the piecewise function isn't discontinuous? Or that the piecewise function is continuous?

And what about my math? Am I going about this the correct way?

 

[Mark44]

I think I understand. So, would it make more sense for me to say that the piecewise function isn't discontinuous? Or that the piecewise function is continuous?

Right, either way. The function is continuous on the entire real line.

And what about my math? Am I going about this the correct way?

Looks OK to me.

 

[epenguin]

For more clarity I would see whether you could simplify the second function - might bring new insight.

 

[Mark44]

For more clarity I would see whether you could simplify the second function - might bring new insight.

It doesn't in this problem, but the OP might or might not have done this. The second function (the rational expression) is continuous at each point on the interval [0, 3]. @grey2q, did you consider this?

 

[grey2q]

It doesn't in this problem, but the OP might or might not have done this. The second function (the rational expression) is continuous at each point on the interval [0, 3]. @grey2q, did you consider this?

I did.

Thanks for the help guys. I think I understand now.