Question

How do you determine removable discontinuity for a function?

Answer 1

Recall that a function `f(x)` is continuous at `a` if

`lim_{x to a}f(x)=f(a)`,

which can be divided into three conditions:

C1: `lim_{x to a }f(x)` exists.
C2: `f(a)` is defined.
C3: C1 = C2

A removable discontinuity occurs when C1 is satisfied, but at least one of C2 or C3 is violated. For example, `f(x)={x^2-1}/{x-1}` has a removable discontinuity at `x=1` since

#lim_{x to 1}{x^2-1}/{x-1} =lim_{x to 1}{(x+1)(x-1)}/{x-1} =lim_{x to 1}(x+1)=2#,

but `f(1)` is undefined.