Wolfram Function Repository
Instantuse addon functions for the Wolfram Language
Function Repository Resource:
Compute the discontinuities of a function of a single variable
ResourceFunction["FunctionDiscontinuities"][f,x] computes the values of x at which f(x) is discontinuous with respect to x. 

ResourceFunction["FunctionDiscontinuities"][{f,cond},x] computes the x for which cond is True and f(x) is discontinuous with respect to x. 

ResourceFunction["FunctionDiscontinuities"][…,x,"Properties"] computes points of discontinuity along with related information about each. 
Compute the points of discontinuity of a rational function:
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Repeat the calculation, classifying the points of discontinuity:
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Compute the points of discontinuity of a trigonometric function:
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Repeat the calculation, classifying the points of discontinuity:
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Compute the points of discontinuity of a smooth function:
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Compute and plot the points of discontinuity of a rational function:
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Compute and plot the points of discontinuity of a step function:
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Compute and plot the points of discontinuity of a step function:
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Choose whether to exclude removable singularities:
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FunctionDiscontinuities has the attribute HoldFirst, enabling calculations such as the following:
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Points where a function approaches ±∞ are considered to be points of discontinuity, even if they are technically outside the range of function definition:
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