I understand that rational functions of certain forms will produce a "hole" or infinitismal break in the graph, like the following:

will obviously appear as the graph of

with a hole at

. But is there a way to explicitly define a function (supposedly composed of functions which are not defined for all values of x, or that may result in such a function) to explicitly define a function [without using domain restrictions, piecewise representations, or band-aid set theory definitions; (i.e. using set theory to to define the function with the desire discountinuities)] that would naturally have discountinuities over a specific interval? Say I wanted to explicitly define a function that had a complete discontinuity at all values

or more generally

is there a manner in which it is possible to construct a function that would produce this property?

Recap:

[1]Explicit, non-piece-wise function

. No using set theory directly "cut and delete". No given domain restrictions

[2]

is not simply discontinuos at specific values, or holes, of

, but rather is discontinuos for

How would one go about creating such a function? Thanks in advance for any advice related to this topic