Question

What is a discontinuous function?

Answer 1

A discontinuous function is a function with at least one point where it fails to be continuous.

That is `lim_(x->a) f(x)` either does not exist or is not equal to `f(a)`.

Explanation:

An example of a function with a simple, removable, discontinuity would be:

`z(x) = { (1, if x = 0), (0, if x != 0) :}`

An example of a pathologically discontinuous function from `RR` to `RR` would be:

`r(x) = { (1, "if x is rational"), (0, "if x is irrational") :}`

This is discontinuous at every point.

Consider the function

`q(x) = { (1, "if x = 0"), (1/q, "if x = p/q for integers p, q in lowest terms"), (0, "if x is irrational") :}`

Then `q(x)` is continuous at every irrational number and discontinuous at every rational number.