# FunctionDiscontinuities 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

open allclose all

## 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 :

Find the discontinuity points between and :

Visualize the discontinuities:

Find the discontinuities of :

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 :

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: 