Definition:Uniform Continuity

Also see

 * Continuous Real Function
 * Continuous Mapping on a Metric Space


 * Absolute Continuity

Relationship to Continuity
The property that $f$ is uniformly continuous on $I$ is stronger than that of being continuous on $I$.

Intuitively, continuity on an interval means that for each fixed point $x$ of the interval, the value of $\map f \y$ is near $\map f x$ whenever $y$ is close to $x$.

But how close you need to be in order for $\size {\map f x - \map f y}$ to be less than a given number may depend on the point $x$ you pick on the interval.

Uniform continuity on an interval means that this can be chosen in a way which is independent of the particular point $x$.

See the proof of this fact for a more precise explanation.

Relationship to Absolute Continuity
The property that $f$ is uniformly continuous on $I$ is weaker than the property that $f$ is absolutely continuous on $I$.

That is, absolute continuity implies uniform continuity.

Compare

 * The difference between convergence and uniform convergence.

Also see

 * Uniformly Continuous Function Preserves Uniform Convergence