Continuity and Uniform Continuity
Continuity as a sequence or epsilon-delta condition, then the stronger version where one delta works across the whole domain. Heine-Cantor says a continuous function on a closed bounded interval is uniformly continuous, and the extreme value theorem says it attains its bounds.
What a learner can do afterwards
- Prove continuity of a function at a point from the definition
- Give a continuous function that is not uniformly continuous, with the reason
- State the extreme value theorem and where its hypotheses are used
- Use closed-interval compactness to explain why continuity implies uniform continuity.
1 · Read
A function is continuous at a point c when its value exists, its limit exists, and the two are equal. The epsilon-delta form says that for every positive epsilon there is a positive delta such that |x minus c| below delta forces |f(x) minus f(c)| below epsilon. Equivalently, every sequence converging to c produces outputs converging to f(c). For f(x) equal to 3x at 2, delta equal to epsilon over 3 works. Most proofs chase an arbitrary tolerance back to such a distance. A sequence scene can show the same idea: x_k moves toward c while f(x_k) moves toward f(c).
Uniform continuity uses one delta for every domain pair: if |x minus y| is below delta, then |f(x) minus f(y)| is below epsilon. The function e to the x on real numbers is continuous, but not uniformly continuous: far right, small input gaps create large output gaps, so no shared delta works. Pointwise continuity may choose a fresh delta at each point.
The extreme value theorem says a continuous function on a closed bounded interval [a, b] attains highest and lowest values. Points x_max and x_min in [a, b] attain them. The same hypotheses invoke Heine-Cantor: a continuous function there is uniformly continuous, so one delta works for every pair. Continuity prevents wild jumps, while closedness keeps endpoint and limit points inside. Drop closedness and it breaks: f(x) equal to x on (0, 1) attains neither bound.
A bounded-slope example separates the ideas. Suppose a function g on all real numbers satisfies |g(x) minus g(y)| at most 2 times |x minus y|. For every epsilon, delta equal to epsilon over 2 works everywhere, so g is uniformly continuous.
A continuous function can fail uniform continuity on an unbounded domain. For f(x) equal to x cubed on all real numbers, the output gap factors as (x minus y)(x squared plus xy plus y squared), and the second factor grows without bound. No single positive delta controls every pair.
Continuity matches limits to values point by point, uniformity shares one delta everywhere, and closed intervals let continuous functions attain their bounds.
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2 · Watch
به انگلیسیTake it off screen
Where it sits
Optional connections
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.