A removable discontinuity occurs in a function when a particular point is undefined or does not match the rest of the function, but the limit of the function exists at that point. This means that the function can be made continuous by redefining it at that specific point.
For example, consider the function:
f(x) = (x^2 - 1) / (x - 1)
At x = 1, the function is undefined because the denominator becomes zero. However, if we factor the numerator, we get:
f(x) = (x - 1)(x + 1) / (x - 1)
We can simplify this to:
f(x) = x + 1
for all x ≠ 1. The limit as x approaches 1 is 2. Therefore, if we redefine f(1) = 2, the function becomes continuous at x = 1. This is an example of a removable discontinuity.