How do I determine the type of discontinuity for a discontinuous function?

[LostMathStudent]

I need help with discontinuous functions. More specifically, how to determine what type of discontinuity they are, algebraically.

Example: Determine whether each function is continuous at the given x values. Justify using the continuity test. If discontinuous, identify the type of discontinuity

(x^2 - 36)/(x+6); at x=-6 and x=6

Alright, so I know the function is continuous at 6, but discontinuous at 6 because the value makes the denominator a zero.

Now, how do I determine what type of discontinuity this function is? Is there a certain way I should preferably be doing this? I'm honestly pretty lost.

 

[mathbalarka]

Note that

$$\frac{x^2 - 36}{x+6} = x-6$$

What can you say about the discontinuities of this function then?

 

[I like Serena]

I need help with discontinuous functions. More specifically, how to determine what type of discontinuity they are, algebraically.

Example: Determine whether each function is continuous at the given x values. Justify using the continuity test. If discontinuous, identify the type of discontinuity

(x^2 - 36)/(x+6); at x=-6 and x=6

Alright, so I know the function is continuous at 6, but discontinuous at 6 because the value makes the denominator a zero.

Now, how do I determine what type of discontinuity this function is? Is there a certain way I should preferably be doing this? I'm honestly pretty lost.

Hi LostMathStudent! Welcome to MHB! :)

We usually distinguish 3 types of discontinuities: removable, jump, and essential.
You can find the classification of discontinuities on wikipedia.
It's possible though that in your textbook the names are slightly different.

In short, it's removable if the function is smooth, but there is just a point missing.
It's a jump if the function makes a jump at the point in question.
And otherwise it is called (by some sources) an essential discontinuity.

What type do you think you have?

 

[LostMathStudent]

Hi LostMathStudent! Welcome to MHB! :)

We usually distinguish 3 types of discontinuities: removable, jump, and essential.
You can find the classification of discontinuities on wikipedia.
It's possible though that in your textbook the names are slightly different.

In short, it's removable if the function is smooth, but there is just a point missing.
It's a jump if the function makes a jump at the point in question.
And otherwise it is called (by some sources) an essential discontinuity.

What type do you think you have?

I know the differences between all the discontinuities, except in our class we call essential discontinuity infinite discontinuity.

Could I factor the function, and then if they cancel out I know I have removable discontinuity? Then, if they don't cancel, I still have essential discontinuity.

I'm not sure if this works on all functions, including square root functions and piecewise functions. I'm assuming that square root functions can only have removable discontinuity, and piecewise can only have jump discontinuity.

 

[I like Serena]

I know the differences between all the discontinuities, except in our class we call essential discontinuity infinite discontinuity.

Could I factor the function, and then if they cancel out I know I have removable discontinuity? Then, if they don't cancel, I still have essential discontinuity.

I'm not sure if this works on all functions, including square root functions and piecewise functions. I'm assuming that square root functions can only have removable discontinuity, and piecewise can only have jump discontinuity.

That sounds pretty good.
Formally these classifications are done based on left-sided and right-sided limits.
Are you familiar with those?
Otherwise your classifications seem to hit the mark.

However, suppose you have the piecewise function:
$$f(x)=\begin{cases}
x &\text{if }x \ne 0 \\
1 &\text{if }x = 0 \end{cases}$$
Then the discontinuity at $x=0$ is removable instead of a jump discontinuity.Anyway, in your current problem you can indeed factor and cancel the denominator, making it a removable discontinuity.