##  [Continuity](/continuity-0) 

 Definition

Property of a function at a point (or on a set) that arbitrarily small changes in the input produce arbitrarily small changes in the output; formally, in a metric context: for each point x0 and each ε>0 there exists δ>0 such that d(x,x0)<δ implies d(f(x),f(x0))<ε.

 

 

 

 

 

 





## Principle

Principle

Local control of outputs by inputs: near each point the function's variation can be bounded by choosing a sufficiently small neighborhood determined by the point and the desired output tolerance.

 

 

 

 

 





## Demonstration

Demonstration

The sine function on R is continuous everywhere: for any x0 and ε>0 one can find δ depending on x0 and ε so that |sin x - sin x0|<ε whenever |x-x0|<δ. The Heaviside step function is discontinuous at 0.

 

 

 

 

## Misapplication

Misapplication

Treating continuity as if it guaranteed uniform smallness of variation over the whole domain (confusing pointwise continuity with uniform continuity), or assuming continuity implies boundedness on noncompact domains.

 

 

 

 

 





## Consequence

Consequence

Continuous functions preserve limits and connectedness, and on compact domains they are bounded and attain maxima and minima. Continuity is preserved under sums, products, and composition.

 

 

 

 

## Reversal

Reversal

A discontinuous function has at least one point where arbitrarily small input changes can produce a jump or other nonvanishing change in output; discontinuity is the negation of the local ε–δ property.

 

 

 

 

 





## Boundary

Boundary

Requires a topology or metric to compare 'small' changes; statements about continuity must specify domain and codomain (metric, topological or uniform structure). Continuity as used here excludes weaker, measure-theoretic notions like almost-everywhere continuity unless stated.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often confused with uniform continuity and with differentiability: continuity is strictly weaker than either, and the same word is used for both pointwise and global variants, which creates ambiguity without further qualification.

 

 

 

 

 





## Synthesis

Synthesis

Continuity is the local ε–δ condition guaranteeing that outputs vary arbitrarily little when inputs are restricted to a small neighbourhood of a point; it is the basic continuity of behaviour on which stronger global or smoothness properties build.