continuity?

Question

1. What is a removable discontinuity, and how is it removed?

2. Describe, with an example, where a rational function can be discontinuous.

Answer 1

I think an example of a function with a removable discontinuity is f(x) = x/x. This function is equivalent to f(x) = 1, except it is undefined for x=0, making the function disconuous at that point. To remove the disconuity, you can specify f(0) = 1 for that special case. (I am not 100% sure about this however - don't get mad if your textbook's definition is something different. :) )

Answer 2

f(x) =
x+1, for x>2
x-1, for x<0

If you draw a graphic of this function, you will notice that the function is discontinuous - meaning that you have no values certain x, or that the graphic is not a continuous line, but it has some discontinuouties. Sorry if I wasn't clear enough.
For your 1st question, I'm not too sure, you could remove the discontinuouty by modifying the conditions, here you change from x<0 to x<=2...

Answer 3

for all x for all x>=n, y = 4

the discontinuity is at n ....

1/x is discontinious at x=0

Answer 4

f(x) = 1 for x< 1
f(x) = 2 for x>=1

a discontinuity exists at x = 1 this one can not be reomoved : meaning you can not define f(1) in such a way that f is continous.