What are non-removable discontinuities?
jumps, asymptotes
Non-removable discontinuities are points on a function where the function does not exist or is undefined due to a division by zero. These points are also known as infinite or essential discontinuities, as they cannot be removed by simply modifying or redefining the function at these points.
Non-removable discontinuities occur when the denominator of a fraction becomes zero, leading to an undefined value. For example, the function f(x) = (x-3)/(x^2-9) has a non-removable discontinuity at x = 3 and x = -3, as these values would make the denominator zero.
It is important to note that a non-removable discontinuity is different from a removable discontinuity or a hole in the graph of the function. A removable discontinuity occurs when a point is removed from the graph of a function, but the function can be filled in at that point to make it continuous. This is typically done by redefining the function or using limits to find the missing value at that point.
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