Math 2.0

Real functions Definition

Continuity at a point

Builds on Limit of a function

Definition

Let D⊆ℝD \subseteq \mathbb{R}, let f ⁣:D→ℝf \colon D \to \mathbb{R} be a function, and let a∈Da \in D. The function ff is continuous at aa if the following holds. For each open interval VV containing f(a)f(a) there is an open interval UU containing aa with f(x)∈Vf(x) \in V for all x∈U∩Dx \in U \cap D.

Remarks

If aa is a limit point of DD, then ff is continuous at aa if and only if lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a). See limit of a function. If aa is an isolated point of DD, then each function on DD is continuous at aa. The epsilon-delta form states the same condition with numbers. Mathlib defines it for topological spaces, as ContinuousAt.

Reading path

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinition
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  1. Limit point of a set definition
  2. Limit of a function definition
  3. Continuity at a point definition

Where this leads

250 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Continuous_function, Definition in terms of neighborhoods
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