functions - Testing for continuity over a given domain

Is there any way to use Mathematica to test whether a function is continuous over a given domain?

Answer

This is likely to be very slow..

isDiscontinuous[f_, low_, high_] := 
 Resolve[Exists[del, del > 0, ForAll[eps, eps > 0,
    Exists[{x1, x2}, 
     low <= x1 < x2 <= high && x2 - x1 < eps && 
      Abs[f[x1] - f[x2]] > del]]]]

Here are simple examples.

ff[x_] := x^2 + x

isDiscontinuous[ff, -1, 2]

(* Out[334]= False *)

gg[x_] := Sign[x]

Resolve[isDiscontinuous[gg, -1, 2]]


(* Out[331]= True *)

Comments

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