Math 2.0

definition

Continuous function

Definition

Let D⊆RD \subseteq \mathbb{R} and let f ⁣:D→Rf \colon D \to \mathbb{R} be a function. The function ff is continuous if it is continuous at each point of DD. For a subset S⊆DS \subseteq D, ff is continuous on SS if its restriction to SS is continuous.

Remarks

Sums, products, quotients, and compositions of continuous functions are continuous. Polynomial functions are continuous. So are the exponential function, the sine, and the cosine. On a closed bounded interval, a continuous function has the intermediate value property and attains its extreme values. Mathlib defines it for topological spaces, as Continuous.

Explanations

L1 human

A continuous function has no jumps or breaks at any point of its domain. On an interval, we can draw its graph without lifting the pen.

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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 pointdefinitionContinuous function, definition, no evidence levelContinuous functiondefinition
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  1. Limit point of a set definition
  2. Limit of a function definition
  3. Continuity at a point definition
  4. Continuous function definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Continuous_function, Definition

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