FunctionDiscontinuities

FunctionDiscontinuities[f,x]

finds the discontinuities of for xReals.

FunctionDiscontinuities[f,x,dom]

finds the discontinuities of for xdom.

FunctionDiscontinuities[{f1,f2,},{x1,x2,},dom]

finds the discontinuities of for x1,x2,dom.

Details

  • Function discontinuities are typically used to either find regions where a function is guaranteed to be continuous or to find points and curves where special analysis needs to be performed.
  • FunctionDiscontinuities gives an implicit description of a set such that is continuous in . The set is not guaranteed to be minimal.
  • The resulting implicit description consists of equations, inequalities, domain specifications and logical combinations of these suitable for use in functions such as Reduce and Solve etc.
  • Possible values for dom are Reals and Complexes.

Examples

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Basic Examples  (4)

Find the discontinuities of a real univariate function:

Find the discontinuities of a complex univariate function:

Find the discontinuities of a real multivariate function:

Find the discontinuities of a complex multivariate function:

Scope  (5)

Discontinuities of a real univariate function:

Find the discontinuity points between and :

Visualize the discontinuities:

Discontinuities of a function composition:

Find a finite set of discontinuity points between and :

Visualize the discontinuities:

Discontinuities over the reals include the points where the function is not real valued:

Discontinuities of a complex univariate function:

Compute the discontinuities in terms of Re[z] and Im[z]:

Visualize the discontinuities:

Discontinuities of a real multivariate function:

Visualize the discontinuities:

Applications  (6)

Basic Applications  (4)

Find the discontinuities of x+cos(x) TemplateBox[{{{sin, (, x, )}}}, UnitStepSeq]:

Find the discontinuity points between and :

Visualize the discontinuities:

Find the discontinuities of max(log(TemplateBox[{x}, Abs]+1),x sin(x)):

Show that there are no discontinuities:

The function is continuous:

Find the discontinuities of the complex function :

Compute the discontinuities in terms of Re[z] and Im[z]:

Visualize the discontinuities:

Find the discontinuities of given the discontinuities of and :

Suppose the discontinuities of and are contained in solution sets of and :

The discontinuities of are contained in the solution set of :

Calculus  (1)

If is continuous at , then TemplateBox[{{f, (, x, )}, x, a}, Limit2Arg, DisplayFunction -> ({Sequence[{Sequence["lim"], _, DocumentationBuild`Utils`Private`Parenth[{#2, ->, #3}, LimitsPositioning -> True]}], #1} & ), InterpretationFunction -> ({Limit, [, {#1, ,, {#2, ->, #3}}, ]} & )]=f(a):

Check that is continuous at :

The limit of at can be found by simple substitution:

Visualization  (1)

Use discontinuities to find Exclusions settings for Plot:

Convert the discontinuities into the format required by Exclusions:

Use the exclusions in Plot:

Compare to a plot computed without using exclusions:

Properties & Relations  (2)

The function is continuous outside the set given by FunctionDiscontinuities:

Use FunctionContinuous to check the continuity:

FunctionSingularities gives a set outside which the function is analytic:

The set of discontinuities is a subset of the set of singularities:

Possible Issues  (2)

The discontinuity set returned may not be minimal:

The function is identically zero, hence it has no discontinuities:

When some discontinuity information is missing, an error message is given and the known discontinuities are returned:

Wolfram Research (2020), FunctionDiscontinuities, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionDiscontinuities.html.

Text

Wolfram Research (2020), FunctionDiscontinuities, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionDiscontinuities.html.

BibTeX

@misc{reference.wolfram_2020_functiondiscontinuities, author="Wolfram Research", title="{FunctionDiscontinuities}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionDiscontinuities.html}", note=[Accessed: 17-April-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_functiondiscontinuities, organization={Wolfram Research}, title={FunctionDiscontinuities}, year={2020}, url={https://reference.wolfram.com/language/ref/FunctionDiscontinuities.html}, note=[Accessed: 17-April-2021 ]}

CMS

Wolfram Language. 2020. "FunctionDiscontinuities." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionDiscontinuities.html.

APA

Wolfram Language. (2020). FunctionDiscontinuities. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionDiscontinuities.html