Calculus I · Limits and Continuity · lesson
Continuity at a Point
Learn Continuity at a Point with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.
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 at a Point; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Complete Rational Function Asymptote Analysis to Continuity on Intervals and at Endpoints. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Use the three-part continuity test at a point; distinguish continuity from merely having a limit; explain the graphical meaning of continuity.
Continuity and the Intermediate Value Theorem
Continuity at a Point
A limit describes nearby behavior. Continuity connects that nearby behavior to the function's actual value.
Continuity at \(x=a\)
A function is continuous at if all three conditions hold:
• is defined; • exists; • .
Equivalently,
Continuity means the graph, the nearby trend, and the actual dot all agree at the point. You approach one height, and the function is actually located at that height.
A continuous line
Is continuous at ?
Show worked solution
Check the three conditions.
1. The function value exists:
2. The limit exists:
3. They agree:
Therefore, is continuous at .
A limit exists but continuity fails
Let
Is continuous at ?
Show worked solution
Condition 1: , so the function value exists.
Condition 2: Nearby values use , so
The limit exists.
Condition 3: Compare:
Therefore, is not continuous at .
This discontinuity is removable. Redefining would repair it.
Continuity fails because no two-sided limit exists
Let
Is continuous at ?
Show worked solution
The function value exists:
But
while
The one-sided limits disagree, so the two-sided limit does not exist. Condition 2 fails. Therefore, is not continuous at .
Do not use only the slogan "draw without lifting your pencil." It is a helpful picture, not a complete test. At endpoints, one-sided continuity is allowed; on disconnected domains, a graph can be continuous at every point of its domain even though you cannot draw all pieces in one stroke.
Source & rights
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