For your second question: consider an arbitrary meromorphic on the complex plane with poles at points z1, z2, ...,

Now consider the same function, but now restricted to C \ {z1,z2,...}. By restricting the domain, the function is now complex analytic. Suddenly you went from having singularities to as smooth as possible.

Consider also Lusin's theorem, which implies the following Corollary: given an absolutely continuous function f defined on the interval [0,1], for every positive epsilon, there is a subset E of measure less than epsilon, such that f is C^1 on [0,1] \ E.

]]>'things like probability distributions are by-design a type of function "with singularities"' I am not able to understand this statement, to my understanding probability distributions are functions like the Gaussian distribution, Rayleigh distribution, etc., and they need not contain any singularities.

Also I do not understand the meaning of 'One of the key problems with "functions with singularities" is you can usually just change the domain of the function, to get a smooth function on another domain'.

request you to kindly clarify my doubts.]]>

One of the key problems with "functions with singularities" is you can usually just change the domain of the function, to get a smooth function on another domain (now with its own singularities, in a sense). This is the kind of thing we're doing (more or less) when we talk about piecewise C^1 or piecewise analytic functions. And these come up naturally enough, for example in a collision between two bodies that involves friction. So although you can approximate by smooth functions, it's in no way "natural" to the model.

Of course there's a very simple answer to your question -- things like probability distributions are by-design a type of function "with singularities".]]>