Exploring Types of Discontinuities AB & BC
A function can fail to be continuous at a point in three ways, distinguished by the one-sided limits. If both sides agree but the function value is missing or different, the discontinuity is removable, a hole. If both sides are finite but unequal, it is a jump. If either side is unbounded, it is infinite, and the graph has a vertical asymptote. The classification is made from the limits, never from how the formula looks.
Two errors dominate. The types get conflated, so a jump is called removable because the function is defined on both sides, or a hole is called a jump because the graph is broken. And every zero of a denominator gets read as a vertical asymptote, when a factor that cancels against the numerator produces a hole instead. Factoring completely and looking at what survives in the reduced denominator is what separates the two, and skipping it turns the classification into a guess.
The work
3 ways in · any order
Lesson
Exploring Types of Discontinuities
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Classifies the three discontinuity types by their one-sided limits, shows how to find each from a factored formula, and drills the difference between a cancelling and a surviving denominator factor.
Diagnostic
10-item topic check
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Ten items spanning the two failure modes of this topic: conflating the discontinuity types, and calling every denominator zero an asymptote. Take it cold to find which one is yours, or after the lesson to confirm it is not.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.