I have just used DBSCAN to detect the outliers. If $\epsilon$ is the radius as the given parameter, then a point is outlier means it's distance with all points greater than $\epsilon$, But I have just checked that each points has pair distance less than $\epsilon$. So none of the data should be outliers. But DBSCAN has returned some outliers. Is it theoretically ok! Or I'm making bad mistake. Thanks for reading.
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If DBSCAN says a point is an outlier, it does not mean that its distance with all points is greater than ϵ. DBSCAN has two parameters, ϵ and n. If a point is an outlier, then it will have fewer than n points within a radius of ϵ (although that is not sufficient to guarantee that the point is an outlier). It would be unusual to set n at 1 (in the R library, the default value of n is 5).