Calculus I · Limits and Continuity · lesson
Continuity on Intervals and at Endpoints
Learn Continuity on Intervals and at Endpoints with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style pr
Where this chapter fits
Chapter 5: Continuity
Connect limits to function values, classify discontinuities, repair piecewise definitions, and use the Intermediate Value Theorem.
Reading lens: Do the limit, the function value, and the surrounding domain fit together at the point or across the interval? Keep that question in view while reading Continuity on Intervals and at Endpoints; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Continuity at a Point to Which Functions Are Continuous?. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Determine intervals of continuity; use one-sided limits at endpoints; state the natural domains on which familiar functions are continuous.
Continuity on Intervals and at Endpoints
A function is continuous on an open interval if it is continuous at every point in the interval.
At the left endpoint of a closed interval , continuity requires
At the right endpoint , continuity requires
At the left edge of a road, there is no road to the left. It would be silly to demand traffic from a direction that does not exist. Endpoint continuity checks only from inside the interval.
Square root at its endpoint
Is continuous at on its domain ?
Show worked solution
The function value is
Only the right-hand limit is relevant because negative inputs are outside the real domain:
The right-hand limit equals the function value, so is continuous at the endpoint .
Source & rights
Original instruction with traceable references.
The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.
The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.