What is a Removable Discontinuity?
A removable discontinuity has a gap that can easily be filled in, because the limit is the same on both sides. You can think of it as a small hole in the x-axis.
A removable discontinuity is sometimes called a point discontinuity, because the function isn’t defined at a single (miniscule point).
Removing The Hole
The hole is called a removable discontinuity because it can be filled in, or removed, with a little redefining of the function’s values. Simply replace the function value at the hole with the value of the limit.
Take the following piecewise function:
Graphed, the function looks like this:
Note the small hole at x = 0.5. The function value here (i.e. the y-value) is 4, creating a problem if we want to perform further calculations on the function (like integration, for example).
On the graph, you can just pencil the hole in and remove the dot at (0.5, 4). Mathematically, if we take the y-values very close to the hole, we can fill it in that way. The piecewise function is given as h(x) = 1.5 + 1 / (x + .25) for every point except 0.5, so we can ignore that quirk and simply use the function to fill in the hole. Substituting x = 0.5 into the function, we get:
h(x) = 1.5 + 1 / (0.5 + .25) = 17/6 ≈ 2.83.
Bogley, W. (1996). Removable Discontinuities. Retrieved October 28, 2019 from: https://oregonstate.edu/instruct/mth251/cq/Stage4/Lesson/removable.html
Stephanie Glen. "Removable Discontinuity (Point): Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/removable-discontinuity-definition-examples/
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