What is a removable discontinuity?

A removable discontinuity, also known as a hole, is a type of discontinuity in a function where the function is undefined at a specific point, but the limit of the function as x approaches that point exists. In essence, the discontinuity can be "removed" by redefining the function at that single point.

Here's a breakdown of its key characteristics:

• Undefined at a point: The function is not defined at the point of discontinuity (let's call it x = a). This means you cannot directly substitute a into the function's formula to get a y-value.

• Limit exists: The limit of the function as x approaches a exists. This means that the function approaches a specific value from both the left and the right of a. We can write this as: lim_x→a f(x) = L, where L is a finite number.

• Removable by redefinition: The discontinuity can be removed by simply defining the function at x = a to be equal to the limit L. This creates a continuous function over the entire domain, except potentially at other discontinuities the function may possess.

How it arises:

Removable discontinuities often occur due to factors like:

• Cancellation of common factors: A common factor in the numerator and denominator of a rational function can lead to a hole. After canceling the common factor, the resulting simplified function gives the value of the limit.

• Piecewise functions with a gap: A piecewise function may have a gap where the pieces don't meet, but the limit exists.

Example:

Consider the function:

f(x) = (x² - 4) / (x - 2)

This function is undefined at x = 2 because it leads to division by zero. However, we can factor the numerator:

f(x) = (x - 2)(x + 2) / (x - 2)

For x ≠ 2, we can cancel the (x - 2) terms:

f(x) = x + 2

The limit as x approaches 2 is:

lim_x→2 f(x) = 2 + 2 = 4

Thus, there's a removable discontinuity at x = 2. By redefining the function as:

g(x) = x + 2 for all x

we've removed the discontinuity. g(x) is now continuous everywhere.

In summary, a removable discontinuity is a type of discontinuity that's easily fixable by redefining the function's value at a single point, making it a relatively "mild" form of discontinuity compared to jump discontinuities or infinite discontinuities.