Work through the list without notes or a calculator. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. This topic is shared by Calculus AB and Calculus BC.
Understanding
- I can state the three conditions for f to be continuous at x = c, in order. (Guide: “The three-condition definition”; Practice Q1)
- I can explain why f(c) existing, or the limit existing, is not enough on its own. (Guide: “Common misconceptions”; Practice Q1, Q3)
- I can match a hole with no point, a jump and a misplaced point to the condition each one breaks. (Guide: Figure 1; Practice Q3, Q4)
- I can explain why an infinite limit fails condition 2. (Guide: “The three-condition definition”; Practice Q4)
Skills
- I can find f(c) for a piecewise function by choosing the piece whose inequality includes c. (Guide: Worked example 1; Practice Q2, Q5)
- I can find both one-sided limits where a piecewise function changes formula and decide whether the limit exists. (Guide: Worked example 1; Practice Q2, Q5)
- I can find a limit at c by factoring or using a conjugate, then compare it with a separately given f(c). (Guide: Worked example 2; Practice Q1, Q6)
- I can use the sign of x − c to simplify an absolute value on each side of c. (Guide: Worked example 3; Practice Q4(C), Q7)
Reasoning
- I can write a justification that quotes f(c), the limit and the comparison, using the definition. (Guide: “How to write a justification”; Practice Q5, Q6(b), Q7(c))
- I can name the first condition that fails when a function is not continuous at a point. (Guide: “How to write a justification”; Practice Q5(c), Q6(d))
- I can explain why a 0/0 formula does not by itself make a function discontinuous. (Guide: Worked example 2; Practice Q6(c))
- I can explain why no choice of f(c) can make f continuous at c when the limit does not exist. (Guide: Worked example 3; Practice Q7(d))
All ticked? Move on to Topic 1.12, Confirming Continuity over an Interval, or return to the Calculus AB roadmap or the Calculus BC roadmap.