FunctionDiscontinuities
finds the discontinuities of 
for x∈Reals.
FunctionDiscontinuities[f,x,dom]
finds the discontinuities of 
for x∈dom.
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
open allclose allBasic Examples (4)
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:
Applications (6)
Basic Applications (4)
![x+cos(x) TemplateBox[{{{sin, (, x, )}}}, UnitStepSeq]](Files/FunctionDiscontinuities.en/12.png "x+cos(x) TemplateBox[{{{sin, (, x, )}}}, UnitStepSeq]"){kind=small}
Find the discontinuity points between 
and 
:
Visualize the discontinuities:
![max(log(TemplateBox[{x}, Abs]+1),x sin(x))](Files/FunctionDiscontinuities.en/15.png "max(log(TemplateBox[{x}, Abs]+1),x sin(x))"){kind=small}
Show that there are no discontinuities:
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)
![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)](Files/FunctionDiscontinuities.en/29.png "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)"){kind=small}
Visualization (1)
Use discontinuities to find Exclusions settings for Plot:
Convert the discontinuities into the format required by Exclusions:
Use the exclusions in Plot:
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:
Text
Wolfram Research (2020), FunctionDiscontinuities, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionDiscontinuities.html.
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