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 $f \left({y}\right)$ is near $f \left({x}\right)$ whenever $y$ is close to $x$. But how close you need to be in order for $\left\lvert{f \left({x}\right) - f \left({y}\right)}\right\rvert$ 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