Calculus I · Limits and Continuity · review
Continuity and IVT Review
Section 5 Summary
• Continuity at requires a defined value, an existing limit, and equality between them. • At endpoints, use the one-sided limit from inside the interval. • Removable discontinuities can be repaired with one value; jumps and asymptotes cannot. • For piecewise continuity, set the left and right expressions equal at the join. • The Intermediate Value Theorem guarantees an attained value, not its exact location or uniqueness. • Bisection refines an IVT interval into a numerical approximation.
After the explanation
Use the section idea
Do the limit, the function value, and the surrounding domain fit together at the point or across the interval?
Continuity is a three-part agreement: the value exists, the two-sided limit exists, and those two quantities are equal.
At a point, test the three conditions in order; on an interval, check the domain and endpoints before invoking any continuity theorem.
A sign change supports the Intermediate Value Theorem only when continuity holds on the entire closed interval, and it does not prove uniqueness.
You are ready to continue when you can classify a break, decide whether one value can repair it, and state every IVT hypothesis aloud.
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