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Martin's excellent insight into the nature of the problem got me thinking if we could do any better than extension Range where Element : IntegerType. Here is one possibility:

public extension Range where Element : Comparable {
 
 @warn_unused_result
 public func intersect(other: Range) -> Range {
 guard endIndex > other.startIndex else {
 return endIndex..<endIndex
 }
 guard other.endIndex > startIndex else {
 return startIndex..<startIndex
 }
 let s = other.startIndex > startIndex ? other.startIndex : startIndex
 let e = other.endIndex < endIndex ? other.endIndex : endIndex
 return s..<e
 }
}

This at least passes Martin's range of string test:

let r1 = "f🇩🇰oo".rangeOfString("🇩🇰")! // 1..<5
let r2 = "ba🇨🇰r".rangeOfString("🇨🇰")! // 2..<6
let r = r1.intersect(r2) // 2..<5