Justify conclusions about continuity at a point using the definition.
Determine intervals over which a function is continuous.
I can determine whether a discontinuity is removable, infinite, or a jump.
I can identify which part of the definition is violated for each kind of discontinuity.
I can use proper interval notation to identify intervals of continuity as all x-values that are not discontinuous.
Quick Lesson Plan
This lesson introduces students to the idea of continuity using the analogy of a blind date. There are many things that can go wrong that would cause the two people to not actually meet at Starbucks (i.e. various types of discontinuity). Students work through various cases and then infer the definition of continuity. The debrief portion adds formal limit notation to each scenario.
Students’ understanding of continuity is pretty intuitive. If they don’t have to pick up their pencil when tracing the graph from left to right, then the function is continuous. They will need more support in attaching the formal definition of continuity to their justifications. We recommend using the two-part definition of continuity (limits from left and right are equal and that the limit equals the y-value) and having students verify both conditions each time they must prove a function is continuous at a certain x-value. Some teachers choose to include a third condition: f(a) exists, but we prefer to embed this condition within the shorter two-part definition.
Sharing the graph of a function with an oscillating discontinuity (like f(x) = sin (1/x)) may be valuable for students, but not tested on the exam.
Although it is not commonly tested, it is worth mentioning that a function can be continuous at its endpoint if the one-sided limit matches the y-value. In CYU #1, we intentionally included endpoints to generate conversation about this. Additionally, we call x=4 an infinite discontinuity, but some textbooks would not mention this point because it is not included in the domain of the function. For consistency, refer to the definition of continuity in your text when discussing x-values not in the domain of a function.
CYU #2 is directly from an old AP Calculus exam (2011, AB, 6a). You could pull up the scoring guidelines to give students a real idea of what kind of justification is necessary. The question was worth two points.