EpsilonDelta.swift

import Foundation

typealias R = Double
let step = 10

func unitRand() -> R {
    R(arc4random_uniform(UINT32_MAX)) / R(UINT32_MAX)
}

func forAnySmall(_ p: (R) -> Bool) -> Bool {
    for i in 0 ..< step {
        let ε = R(step - i) / R(step)
        if !p(ε) {
            return false
        }
    }
    return true
}

func existsSmall(_ p: (R) -> Bool) -> Bool {
    for i in 0 ..< step {
        let δ = R(step - i) / R(step)
        if p(δ) {
            return true
        }
    }
    return false
}

func forAnyInRange(_ from: R, _ to: R, p: ((R) -> Bool)) -> Bool {
    for _ in 0 ..< step {
        let x = from + unitRand() * (to - from)
        if !p(x) {
            return false
        }
    }
    return true
}

func isContinuous(_ f: (R) -> R, at a: R) -> Bool {
    forAnySmall { ε in
        existsSmall { δ in
            forAnyInRange(a - δ, a + δ) { x in
                abs(f(x) - f(a)) < ε
            }
        }
    }
}

print( isContinuous( {x in pow(x, 2)}, at: 0) )
// true: x^2 is continuous at x = 0

print( isContinuous( {x in floor(x)}, at: 1) )
// false: floor func is not continuous at x = 0