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

With the assumption that , FunctionContinuous succeeds:

By default, FunctionContinuous may generate conditions on symbolic parameters:

With GenerateConditionsNone, 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 GenerateConditionsAutomatic, conditions that are generically true are not reported:

Use PerformanceGoal to avoid potentially expensive computations:

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

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:

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:

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: