Sketch the graph of : the limit of f of x as it goes to 1 from the right side is 2. the limit of f of x as it goes to 1 from the left is -1. f of 1 is undefined. & is the function continuous in the interval [-5,5]?

Question

Sketch the graph of : the limit of f of x as it goes to 1 from the right side is 2. the limit of f of x as it goes to 1 from the left is -1. f of 1 is undefined.  & is the function continuous in the interval [-5,5]?

debbieg · Answer

So it will have an open point at (1,2) extending to the positive direction of the x axis (however you want it to really, so long as it is still a function, e.g., passes the vertical line test). Another open point at (1,-1) extending to the left.

To be continuous at any x=a, the limits at a from the left and right must be equal, and must equal f(a). Do you think this function is continuous at x=1?

anonymous · Answer

Well if they're not equal then the limit doesn't exist right? so x=1 isn't continuous.

debbieg · Answer

The 2-sided limit does not exist; right. In order for the 2-sided limit to exist, the 1-sided limits must be equal. So not continuous, because the limit doesn't exist AND because f(1) doesn't exist.

anonymous · Answer

Okay, I understand that part. But what I don't understand is whether the function is continuous in the interval [-5,5)

debbieg · Answer

To be continuous on an interval, a function must be continuous at every point in the interval. Since x=1 is in the interval and we have concluded that there is a discontinuity there..... it is not continuous on that interval.