Question

What does it mean when it says: "non-removable discontinuity at x=4"?

Answer 1

See the explanation.

Explanation:

Here is the definition I am familiar with.

Function `f` has a discontinuity at `a` if `f` is defined in an open interval containing `a` except possibly at `x=a`, and `f` is not continuous at `a`.

(This allows us to avoid saying, for example that `sqrtx` has a discontinuity at `x=-5`, which is good. But, the function `f(x) = 1` with domain all irrational numbers is not continuous at `3`, but is also not discontinuous at `3`, because is it not defined on an open interval containing `3`.)

`f` has a removable discontinuity at `a` if `lim_(xrarra) f(x)` exists. (Remember that writing `lim_(xrarra)f(x)=oo` is a way of explaining why the limit Does Not Exist.)

`f` has a non-removable discontinuity at `a` if `f` has a discontinuity at `a` and `lim_(xrarra) f(x)` does not exist.

In the most familiar functions:

Rational and trigonometric functions have non-removable discontinuities at their vertical asymptotes. (Holes are removable.)

Piecewise-defined functions can have jump discontinuities, which are non-removable. (Holes are removable.)

The Greatest integer function (a.k.a. the Floor function) has a non-removable discontinuity at every integer.