# Is there a functionality to analytically find discontinutites of function?

I would like know whether it is somehow possible to analytically get discontinuities of a given function on a given interval (possibly with some reasonable assumption of the function otherwise being well-behaved)

To give an example I would like a function Discontinuities that behaves something like this:

( *In *) Discontinuities[Sign[x], {x,-1,1}]
(* Out *) {0}

(* In *) Discontinuities[Floor[2x], {x,-1,1}]
(* Out *) {-1/2,0,1/2}

(* In *) Discontinuities[CDF[BinomialDistribution[2, 1/2], x], {x, -1, 3}]
(* Out *) {0,1,2}

• This is a slightly related discussion Aug 19 '20 at 11:55
• Interesting. Something similar could work using the analytical definition of discontinuity. How could I make a subexpression like Limit[Sign[x], x -> 0, Direction -> "FromAbove"] not directly evaluate, so I can Reduce it for x in a logical formula requiring limit "FromAbove" and "FromBelow" to be different at point x? Aug 19 '20 at 12:32
• There's one in the Wolfram Function Repository here. Aug 19 '20 at 13:20
• Specifying the interval in the form {x, -1, 1} is ambiguous as to whether the endpoints should be included. Including intervals as constraints (e.g., {f[x], -1 < x < 1} or {f[x], -1 <= x <= 1}as used in "FunctionDiscontinuities") eliminates the ambiguity. Aug 19 '20 at 15:55