# Dini continuity

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In mathematical analysis, **Dini continuity** is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is Dini continuous.

## Definition

Let be a compact subset of a metric space (such as ), and let be a function from into itself. The modulus of continuity of is

The function is called **Dini-continuous** if

An equivalent condition is that, for any ,

where is the diameter of .

## See also

- Dini test — a condition similar to local Dini continuity implies convergence of a Fourier transform.

## References

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