A removable discontinuity occurs at a point in a function where the function is not defined or does not equal the limit at that point, but can be 'fixed' by redefining the function at that point. This type of discontinuity suggests that while there is an interruption in the function's graph, it could be made continuous by assigning an appropriate value to that specific point. Recognizing removable discontinuities is crucial when analyzing the limits and continuity of functions in multiple variables, as they often provide insights into the behavior of the function around those points.
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Removable discontinuities often arise in rational functions where a factor cancels out in both the numerator and denominator.
To determine if a removable discontinuity exists, check if the limit exists at that point, even if the function does not.
You can find removable discontinuities by identifying points where the function is undefined but has a limit.
Graphically, removable discontinuities appear as holes in the graph of the function.
When redefining a function to remove the discontinuity, it’s essential to set the new value equal to the limit at that point.
Review Questions
How can you identify a removable discontinuity in a multi-variable function?
To identify a removable discontinuity in a multi-variable function, look for points where the function is not defined or does not match its limit. This can occur when evaluating limits from different paths leads to an indeterminate form. If you can find a limit that exists at that point despite the function being undefined there, then you have discovered a removable discontinuity.
Explain how you would go about redefining a function to remove its discontinuity.
To redefine a function to remove its discontinuity, first identify the point where the removable discontinuity occurs. Next, calculate the limit of the function as it approaches that point. You would then assign this limit value to the function at that specific point. This process effectively 'fills in' the hole in the graph, making the function continuous at that location.
Evaluate how understanding removable discontinuities aids in analyzing more complex functions and their behaviors.
Understanding removable discontinuities is essential for analyzing complex functions because it helps clarify how these functions behave around problematic points. Recognizing where these discontinuities occur allows for smoother transitions when studying limits and can guide you in finding derivatives and integrals. By addressing removable discontinuities, one can better understand continuity across different domains and ensure accurate modeling of real-world scenarios.
Related terms
limit: The value that a function approaches as the input approaches a specified value.
continuity: A property of a function where it is uninterrupted and can be drawn without lifting a pencil from the paper.
non-removable discontinuity: A type of discontinuity that cannot be 'fixed' or made continuous by redefining the function at that point, such as a jump or infinite discontinuity.