Function: math[cont] - verifies continuity at a point

Calling Sequence:

cont(f, x=a);

Parameters:

f - an algebraic expression in x
x - the indeterminate (a name)
a -
a point (value of type realcons)

Description:

cont checks for continuity of a function f in x at a point a.

The function only permits the expression to include one variable and checks whether a point is continuous in the real domain.

The function returns either true or false and in many cases gives further detail on the type of continuity or discontinuity. The function even returns false if f is not defined at a. Enough information is returned to determine those cases (see below).

cont evaluates to the following values:

'true'
if f is defined in the vicinity of a and at a and
[Maple Math] .

'true, left'
if f is defined in the left vicinity of a and at a, but not at the right vicinity and
[Maple Math] .

'true, right'
if f is defined in the right vicinity of a and at a, but not at the left vicinity and
[Maple Math] .

'false, defined, pole', `change of sign`=[+/- [Maple Math] , +/- [Maple Math] ]
if f is defined at a, and
if (
[Maple Math] ) or ( [Maple Math] = [Maple Math] and [Maple Math] = - [Maple Math] ).

'false, undefined, pole', `change of sign`=[+/- [Maple Math] , +/- [Maple Math] ]
if f is
not defined at a, and
if (
[Maple Math] = - [Maple Math] and [Maple Math] = [Maple Math] ) or ( [Maple Math] = [Maple Math] and [Maple Math] = - [Maple Math] ).

'false, defined, pole', `no change of sign`=[+/- [Maple Math] , +/- [Maple Math] ]
if f is defined at a, and
if (
[Maple Math] = - [Maple Math] and [Maple Math] = - [Maple Math] ) or ( [Maple Math] = [Maple Math] and [Maple Math] = [Maple Math] ).

'false, undefined, pole', `no change of sign`=[+/- [Maple Math] , +/- [Maple Math] ]
if f is
not defined at a, and
if (
[Maple Math] = - [Maple Math] and [Maple Math] = - [Maple Math] ) or ( [Maple Math] = [Maple Math] and [Maple Math] = [Maple Math] ).

'false, defined, removable=r'
if f is defined at a
and
[Maple Math] = [Maple Math] and if both limits exist (are finite) and
f(a) <>
[Maple Math] .
cont also returns a value r that gives the value with which f(a) can be replaced. r is determined by the right limit.

'false, undefined, removable=r'
if f is
not defined at a
and
[Maple Math] = [Maple Math] and if both limits exist (are finite) .
cont also returns a value r that gives the value with which f(a) can be defined. r is determined by the right limit.

'false, defined, jump', 'amount'=a
if f is defined at a and
if
[Maple Math] and both limits exist (i.e. are finite). The amount a of the jump is also returned as [Maple Math] .

'false, undefined, jump', 'amount'=a
if f is defined at a and
if
[Maple Math] and both limits exist (i.e. are finite). The amount a of the jump is also returned as [Maple Math] .

'false, isolated'
if f is defined at a but not defined in the vicinity of a.

'false, complex'
if a is not included in the real but in the complex domain.

'false'
is returned in all other cases.

cont also accepts piecewise-defined functions.

Internally, cont evaluates f(a), limit(f, x=a, left), limit(f, x=a, right) and compares their values. Furthermore cont checks the nearby vicinity of f(a) for complex values. Functions iscont and discont are not used by the function.

While cont returns further information on the continuity or discontinuity math/isCont just returns true or false.

This function is part of the math package, and so can be used only after performing the command with(math) or with(math, cont).

Examples:

> with(math, cont):

> cont(ln(x), x=0);

[Maple Math]

> cont(ln(x), x=-1);

[Maple Math]

> cont(sqrt(x), x=0);

[Maple Math]

> cont(1/x, x=0);

[Maple Math]

> cont(exp(-1/x^2), x=0);

[Maple Math]

> cont(floor(x), x=0);

[Maple Math]

> f := x -> sqrt(x^3+4*x^2+4*x): # domain x=-2 and x>=0

> cont(f(x), x=-2);

[Maple Math]

> cont(f(x), x=0);

[Maple Math]

> cont(f(x), x=1);

[Maple Math]

See Also:

math/isCont , math/singularity , discont , math/nondiff .

Version History:

Version 1.0 - May 11, 1997

Version 1.1 - January 01, 1998
Version 1.11 - June 14, 1998

Version 1.12 - December 14, 1998
Version 2.0 - December 19, 1998

Version 2.0.1 - January 06, 1999