Continuity fails in three distinguishable ways, and the difference matters because only one of them can be repaired.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Removable
The limit exists but does not equal the function value, or the function is undefined there. The graph has a single hole.
Why it is called removable
Redefining the function at that one point closes the hole. No other kind of break can be repaired this way.
Jump
The one-sided limits both exist but differ, so the graph steps from one height to another. Piecewise functions produce these.
Infinite
The values run away near the point, which is what happens at a vertical asymptote.
How to classify one
Take both one-sided limits. Equal and finite means removable; different and finite means jump; unbounded means infinite.
Removable discontinuities
A hole: the limit exists but the function is undefined or defined elsewhere at that point. Redefining a single value repairs it, which is what "removable" means.
Jump discontinuities
Both one-sided limits exist and differ, so the graph steps. No redefinition can repair it. Piecewise functions whose pieces disagree at the boundary produce these, as do step functions like postage rates.
Infinite discontinuities
The function grows without bound near the point, producing a vertical asymptote. These arise where a denominator vanishes without a matching factor in the numerator.
Factoring identifies which
Factor numerator and denominator. A denominator factor that cancels gives a hole; one that does not gives an asymptote. The algebra distinguishes the cases before you plot anything.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 0.50)
Step 3: Watch an Example
One step at a time.
Watch Sana Classify a Break
Sana examines f(x) = (x² − 4) ÷ (x − 2) at x = 2.
- Step 1
Substituting 2 gives 0 ÷ 0, so the function is undefined there.
Step 4: Your Turn
Practice makes it stick.
The Hole
Problem 1 of 2
f(x) = (x² − 9) ÷ (x − 3). What value at x = 3 would repair the break?
The Step
Problem 2 of 2
Left limit 2, right limit 5. Which kind of discontinuity? Enter 1 removable, 2 jump, 3 infinite.
Name the Break
1 of 8
Left limit 4, right limit 4, function undefined. Which kind? 1, 2 or 3.
2 of 8
Left limit 1, right limit 6. Which kind? 1, 2 or 3.
3 of 8
Values run away near the point. Which kind? 1, 2 or 3.
4 of 8
f(x) = (x² − 25) ÷ (x − 5). What repairs it at x = 5?
5 of 8
f(x) = 1 ÷ (x − 4). Which kind at x = 4? 1, 2 or 3.
6 of 8
How many kinds of discontinuity are named here?
7 of 8
Match each description with its kind of discontinuity.
Tap a card on the left to start.
8 of 8
Which kind can be repaired by redefining one point? 1, 2 or 3.
Step 5: Quick Check
Show what you know.
Question 1 of 2
Left limit 3, right limit 3, function value 8. Which kind? 1, 2 or 3.
Question 2 of 2
Why is a removable discontinuity called removable?
What You Learned
- A removable discontinuity is a single hole and can be repaired.
- A jump has two finite one-sided limits that disagree.
- An infinite discontinuity occurs at a vertical asymptote.