removable discontinuity
In mathematics, a removable discontinuity refers to a point on the graph of a function where the function is undefined or has a hole, but the hole can be filled in to make the function continuous at that point
In mathematics, a removable discontinuity refers to a point on the graph of a function where the function is undefined or has a hole, but the hole can be filled in to make the function continuous at that point. In other words, at a removable discontinuity, the function can be redefined or modified by assigning a value to the point of discontinuity so that the function becomes continuous.
Removable discontinuities occur when a rational function has a factor in both the numerator and the denominator that cancels out, resulting in a hole in the graph. For example, consider the function f(x) = (x^2 – 1)/(x – 1). At x = 1, the function is undefined because division by zero is not defined. However, by factoring the numerator as (x – 1)(x + 1), we see that the function simplifies to f(x) = x + 1 for all x ≠ 1. By assigning the value of 2 to f(1), we fill in the hole at x = 1 and make the function continuous. Therefore, the discontinuity at x = 1 is removable.
To further illustrate this concept, let’s graph the function f(x) = (x^2 – 1)/(x – 1) using a graphing calculator or software:
The graph of f(x) will show a hole at x = 1, where the function is not defined. However, the rest of the graph will show a continuous line.
It is important to note that not all discontinuities are removable. Other types of discontinuities include jump discontinuities and essential discontinuities, where the function cannot be modified to eliminate the discontinuity and make it continuous. Removable discontinuities are characterized by the possibility of removing the hole or assigning a value to the function at the point of discontinuity to fill in the gap and create a continuous function.
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