How to determine if a function is continuous?

Question

I wish to write code for Riemann-Stieltjes integrals in Mathematica.
A necessary condition for the theorem to hold is that the function must be continuous. The domain of the function is a closed real interval containing infinitely many points, so I can't check continuity at each and every point.
I wish to know if there are any built-in functions in Mathematica that determine if a function is continuous or not. Or if there is any function that returns point of singularities of function?

Eduardo Serna · Answer

I think this is impossible, i.e there's no way to know if an arbitrary function is continuous unless you are given some additional information about the function. Otherwise a function is just a black box. If you function is made up of only known functions, you could check for Piecewise and check at the interface between definitions for instance, but this doesn't prove anything

Answered by Eduardo Serna on February 16, 2021

J. M.'s ennui · Answer

The new FunctionContinuous[] function in 12.2 is a step in this direction:
FunctionContinuous[Sin[x], x]
   True

FunctionContinuous[Tan[x], x]
   False

{FunctionContinuous[Sqrt[x], x], FunctionContinuous[{Sqrt[x], 0 < x < ∞}, x]}
   {False, True}

A related function is FunctionDiscontinuities[]:
FunctionDiscontinuities[Tan[x], x]
   Cos[x] == 0

FunctionDiscontinuities[Gamma[x], x]
   Sin[π x] == 0 && x <= 0

How to determine if a function is continuous?

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