# FunctionContinuous

FunctionContinuous[f,x]

tests whether is a real-valued continuous function for xReals.

FunctionContinuous[f,x,dom]

tests whether is a continuous function for xdom.

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

tests whether are continuous functions for x1,x2,dom.

FunctionContinuous[{funs,cons},xvars,dom]

tests whether are continuous functions for xvarsdom restricted by the constraints cons.

# Details and Options • A function is continuous in a set if for all and for all there is a such that for all , implies .
• A function is continuous in a set if for all and for all there is a such that for all , implies .
• If funs contains parameters other than xvars, the result is typically a ConditionalExpression.
• Possible values for dom are Reals and Complexes. The default is Reals.
• If dom is Reals, then all variables, parameters, constants and function values are restricted to be real.
• cons can contain equations, inequalities or logical combinations of these.
• The functions funs need to be defined for all values that satisfy the constraints cons.
• The following options can be given:
•  Assumptions \$Assumptions assumptions on parameters GenerateConditions True whether to generate conditions on parameters PerformanceGoal \$PerformanceGoal whether prioritize speed or quality
• Possible settings for GenerateConditions include:
•  Automatic nongeneric conditions only True all conditions False no conditions None return unevaluated if conditions are needed
• Possible settings for PerformanceGoal are "Speed" and "Quality".

# Examples

open allclose all

## Basic Examples(4)

Test continuity of real functions:

Test continuity of complex functions:

Test continuity over restricted domains:

Test continuity of multivariate functions:

## Scope(6)

Real univariate functions:

Complex univariate functions:

Functions with restricted domains:

Real multivariate functions:

Complex multivariate functions:

Functions with symbolic parameters:

## Options(4)

### Assumptions(1)

FunctionContinuous cannot find the answer for arbitrary values of the parameter :

With the assumption that , FunctionContinuous succeeds:

### GenerateConditions(2)

By default, FunctionContinuous may generate conditions on symbolic parameters:

With , FunctionContinuous fails instead of giving a conditional result:

This returns a conditionally valid result without stating the condition:

By default, all conditions are reported:

With , conditions that are generically true are not reported:

### PerformanceGoal(1)

Use PerformanceGoal to avoid potentially expensive computations:

The default setting uses all available techniques to try to produce a result:

## Applications(14)

### Classes of Continuous Functions(6)

Polynomials are continuous:

Sin, Cos and Exp are continuous:

Visualize these functions:

These functions are continuous in the complex plane as well:

Visualize these functions over :

The reciprocal of a continuous function is continuous wherever :

Thus, rational functions may or may not be continuous over the reals:

However, as every nonconstant polynomial has a root in the plane, rational functions are never continuous on :

Visualizing the function in the complex plane shows the blowup at :

As Cot and Csc are rational functions of Sin and Cos, they are continuous when sine is nonzero:

Visualize the functions along with sine:

Similarly, Tan and Sec are continuous when cosine is nonzero:

This same principle applies to the hyperbolic trigonometric functions Coth and Csch:

Visualize the functions along with Sinh:

As Cosh is never zero, the remaining two functions, Tanh and Sech, are continuous:

The compositions of continuous functions are continuous:

A composition of a discontinuous function and a continuous function will be continuous as long as maps the domain into a continuous subdomain of . Let, for example, be Sqrt. Sqrt is discontinuous on the reals:

However, it is continuous on :

Exp maps :

Thus, the composition of Sqrt with Exp is continuous on :

Multivariate polynomials are continuous over the reals and complexes:

Rational multivariate functions may or may not be continuous over the reals:

They are always discontinuous over the complexes:

Sometimes a discontinuous rational function can be extended to a continuous one:

By composing with continuous univariate functions, many more continuous functions can be generated:

Visualize the continuous functions:

### Calculus(5)

For continuous functions, limits can be computed by substitution:

The functions and agree on the real line except at zero:

Sinc is continuous:

The function is not continuous:

In particular, it is discontinuous at the origin, so its limit there cannot be computed by substitution:

Since the two functions are equal for , they have the same limit there:

The following function is discontinuous:

Its only discontinuity is at the origin:

The discontinuity results from being undefined there:  However, has a limit as :

Define as an extension of to the origin:

This extension is a continuous function:

Visualize :

The function is continuous:

However, its first derivative is not continuous:

Therefore, is not analytic:

While goes smoothly to zero, its derivative oscillates wildly at the origin:

Visualize and its first derivative:

The definite integral of a bounded function is continuous, even if is discontinuous. Consider the following :

It is discontinuous:

Define to be its definite integral from the origin to an arbitrary real value:

The function is continuous

Visualize the function and its integral:

### Probability(3)

The CDF of a continuous probability distribution is continuous:

Visualize the functions:

The CDF of a discrete distribution is discontinuous:

These distributions have piecewise-constant cumulative distribution functions:

The CDF of a mixed distribution is discontinuous:

These distributions have piecewise, but nonconstant, cumulative distribution functions:

## Properties & Relations(3)

At each point of the domain, the limit of a continuous function is equal to its value:

Use Limit to compute limits:

A function continuous in an interval attains each value between its minimum and maximum:

Use Minimize and Maximize to find minima and maxima:

Check that is between the minimum and the maximum:

Use Solve to find the points where attains the value :

Illustrate the property:

Use FunctionAnalytic to check whether a function is analytic:

Analytic functions are continuous:

Continuous functions may not be analytic:

## Possible Issues(2)

A function needs to be defined everywhere to be continuous:

A function needs to be real valued to be continuous over the real domain: