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Everything about Continuous function

Mathematical function with no sudden changes

In mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting its argument to sufficiently small changes. A discontinuous function is a function that is not continuous. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Continuous functionA 5-minute orientation
  2. Skim a recent paper: Generalizing the sigmoid function using continuous-valued logicInfluential work since 2022
  3. Pick up a classic book: Real AnalysisThe most-reprinted book on the subject
  4. Read the landmark paper: The standardisation of terminology of lower urinary tract function: Report from the standardisation sub‐committee of the International Continence SocietyCited 5,731 times
  5. Go sideways: EquicontinuityA closely related topic
  6. Go sideways: Uniform continuityA closely related topic

📖 Definitions

Noun a function whose value at any point in its domain is equal its limit at the same point

Noun a function from one topological space to another, such that the inverse image of any open set is open

🗂️ The subject at a glance

  1. History
  2. Real functions
  3. Continuous functions between metric spaces
  4. Continuous functions between topological spaces
  5. Related notions

🏆 Landmark papers (most cited)

🆕 Influential since 2022

📚 Books worth reading

Ranked by number of editions on Open Library, a rough measure of how lasting a book is.

🔗 Related topics