Question

How do you know when a discontinuity is removable or non-removable? Specifically in an inequality function.

Answer 1

what is an "inequality" function?

Answer 2

such as f(x) = {1/3x +1 when x is less than or equal to 2 and f(x) = 3-x when is x is greater than 2

Answer 3

a discontinuity is removable if you can "remove" it by defining the function in such a way that it becomes continuous
usual example is something like
\[\frac{x^2-9}{x-3}\] which is not continuous at \(x=3\) but would be if you defined it to be equal to 6 when \(x=3\) because that is the limit

Answer 4

oh i see
compute the limit from the left, and the limit from the right, which you do by evaluating both expressons at the point where it changes definition
if you get the same answer, it is continuous
if you don't, it is not

Answer 5

I know there's a discontinuity at 2, but how can i tell if its removable or non-removable?

Answer 6

in your example it is not, because \(\frac{1}{3}\times 2+1\neq 3-2\)

Answer 7

it is not going to be removable if the limits are different
they would have to be equal
in your case they are not, so you have a "jump" discontinuity

Answer 8

removable means basically you have a hole there

Answer 9

which you can remove by filling in the hole

Answer 10

So with these type of problems, when the limits are not equal then the discontinuity cannot be removed? Since if the limits were the same and not different it would be continuous right?

Answer 11

just want to make sure I have the right idea here

Answer 12

yes

Answer 13

in order for the discontinuity to be removable, the limit must exist, only the function for some reason is not defined there
so you have a hole you can fill in
if the right and left hand limits are different, then the actual limit does not exist and you cannot remove the discontinuity