What is a removable discontinuity?

A removable discontinuity occurs in a function when there's a point where the function is not defined, but the limit of the function exists at that point. In simpler terms, it's a "hole" in the graph of the function.

Here's a breakdown:

• Definition: A function f(x) has a <a href="https://www.wikiwhat.page/kavramlar/removable%20discontinuity">removable discontinuity</a> at x = a if:

  1. f(a) is undefined (i.e., a is not in the domain of f).
  2. The <a href="https://www.wikiwhat.page/kavramlar/limit">limit</a> of f(x) as x approaches a exists (i.e., lim (x→a) f(x) = L, where L is a real number).

• How to Identify:

  • Look for factors in the numerator and denominator of a rational function that cancel each other out. The values of x that make these factors equal to zero are potential removable discontinuities.
  • Graphically, a removable discontinuity appears as a single point missing from the graph.

• "Removing" the Discontinuity: You can "remove" the discontinuity by redefining the function at the point x = a such that f(a) equals the limit of f(x) as x approaches a. This effectively "fills in" the hole in the graph. For example, if lim (x→a) f(x) = L, then you can redefine the function as:

  • g(x) = f(x) for x ≠ a
  • g(a) = L

• Examples: Consider the function f(x) = (x² - 4) / (x - 2).

  • f(2) is undefined because the denominator becomes zero.

  • However, we can simplify the function: f(x) = (x - 2)(x + 2) / (x - 2) = x + 2 (for x ≠ 2).

  • The limit of f(x) as x approaches 2 is 4 (since lim (x→2) x + 2 = 4).

  • Therefore, there's a removable discontinuity at x = 2. We can "remove" it by defining a new function:

    • g(x) = (x² - 4) / (x - 2) for x ≠ 2
    • g(2) = 4