FunctionContinuous
FunctionContinuous[f,x]
tests whether is a realvalued continuous function for x∈Reals.
FunctionContinuous[f,x,dom]
tests whether is a continuous function for x∈dom.
FunctionContinuous[{f_{1},f_{2},…},{x_{1},x_{2},…},dom]
tests whether are continuous functions for x_{1},x_{2},…∈dom.
FunctionContinuous[{funs,cons},xvars,dom]
tests whether are continuous functions for xvars∈dom 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 allBasic Examples (4)
Scope (6)
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 GenerateConditionsNone, 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 GenerateConditionsAutomatic, 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)
Sin, Cos and Exp are continuous:
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:
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:
Define as an extension of to the origin:
This extension is a continuous function:
However, its first derivative is not continuous:
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 :
Define to be its definite integral from the origin to an arbitrary real value:
Probability (3)
The CDF of a continuous probability distribution is continuous:
The CDF of a discrete distribution is discontinuous:
These distributions have piecewiseconstant 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 :
Use FunctionAnalytic to check whether a function is analytic:
Text
Wolfram Research (2020), FunctionContinuous, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionContinuous.html.
CMS
Wolfram Language. 2020. "FunctionContinuous." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionContinuous.html.
APA
Wolfram Language. (2020). FunctionContinuous. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionContinuous.html