Originally Posted by

**mfetch22** I understand that rational functions of certain forms will produce a "hole" or infinitismal break in the graph, like the following:

$\displaystyle g(x) = \frac{(x-2)(x+3)}{(x-2)}$

will obviously appear as the graph of $\displaystyle y = x+3$ with a hole at $\displaystyle x=2$. 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

$\displaystyle 2 \leq x \leq 6$

or more generally

$\displaystyle a \leq x \leq b$

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 $\displaystyle f(x)$. No using set theory directly "cut and delete". No given domain restrictions

[2] $\displaystyle f(x)$ is not simply discontinuos at specific values, or holes, of $\displaystyle x$, but rather is discontinuos for

$\displaystyle a \leq x \leq b$

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