This is a practice for identifying and classifying discontinuities of piecewise functions

How You Can (And Should) Get More Practice!

Below is a few practice problems of various difficulty, but you will need considerably more practice than one each. For that reason you should definitely use the green “Try Another” button in the top right corner at least two or three times to complete additional versions of these questions for more practice. You should keep using that button until doing these problems feels straight forward and easy, and then come back after a week or so of doing other stuff and try again to make sure it is still just as easy for you.

Theoretically Easier Difficulty Problem

Start by plugging in the x-value we want to test into both equations to see what values we get.
If one of the functions ends up getting a non-zero over zero (after simplifying), then it has a vertical asymptote there, which means it is not continuous.
If the two values are finite, but not the same number, then that means the y-value jumps, so it is not continuous.
If both numbers are the same, but both inequalities around the x-value of interest are strict inequalities, then the function has no actual value at that x-value, so it is not continuous.
If both numbers are the same, and one of the inequalities has an “or equals to”, then it is continuous!
Consider the following piecewise function:

At the point , the function..

1:
is continuous
2:
is not continuous

Theoretically Medium Difficulty Problem

Start by plugging in the x-value we want to test into both equations to see what values we get.
If one of the functions ends up getting a non-zero over zero (after simplifying), then it has a vertical asymptote there, which means it has an asymptotic discontinuities.
If the two values are finite, but not the same number, then that means the y-value jumps, so it has a jump discontinuity.
If both numbers are the same, but both inequalities around the x-value of interest are strict inequalities, then the function has no actual value at that x-value, so it is a hole.
If both numbers are the same, and one of the inequalities has an “or equals to”, then it is continuous!
Consider the following piecewise function:

At the point , the function..

1:
is continuous
2:
has a hole discontinuity
3:
has a jump discontinuity
4:
has an asymptotic discontinuity

Theoretically Harder Difficulty Problem

Start by plugging in the x-value we want to test into both equations on either side of the x-value of interest to see what values we get. Note that you only need to enter it into two of the subfunctions, not all three.
If one of the functions ends up getting a non-zero over zero (after simplifying), then it has a vertical asymptote there, which means it has an asymptotic discontinuities.
If the two values are finite, but not the same number, then that means the y-value jumps, so it has a jump discontinuity.
If both numbers are the same, but both inequalities around the x-value of interest are strict inequalities, then the function has no actual value at that x-value, so it is a hole.
If both numbers are the same, and one of the inequalities has an “or equals to”, then it is continuous!
Consider the following piecewise function:

At the point , the function..

1:
is continuous
2:
has a hole discontinuity
3:
has a jump discontinuity
4:
has an asymptotic discontinuity