It would help a lot to avoid confusions if the help pages were written in a uniform style: calling conventions, terminology, etc.

Also, I do not see in ?with a mention to subpackages. In particular, showing the difference in syntax between importing a command in a package and a subpackage.

as a mathematical object. Which is its notation? These seems to be the first questions to be answerd then.

I see that there are many definitions around. I find eg:

Mathworld:

 Several notations are commonly used to represent (non-multivalued) functions. The most rigorous notation is , which specifies that is function acting upon a single number (i.e., is a univariate, or one-variable, function) and returning a value . To be even more precise, a notation like ", where " is sometimes used to explicitly specify the domain and range of the function. The slightly different "maps to" notation is sometimes also used when the function is explicitly considered as a "map."

Generally speaking, the symbol refers to the function itself, while refers to the value taken by the function when evaluated at a point . However, especially in more introductory texts, the notation is commonly used to refer to the function itself (as opposed to the value of the function evaluated at ). In this context, the argument is considered to be a dummy variable whose presence indicates that the function takes a single argument (as opposed to , etc.). While this notation is deprecated by professional mathematicians, it is the more familiar one for most nonprofessionals. Therefore, unless indicated otherwise by context, the notation is taken in this work to be a shorthand for the more rigorous .

I disagree here in that most professionals that I know, except  perhaps  mathematicians,  use f(x) to denote a function. Ie x in exp(x)-13 is meant a dummy variable. If this is good or bad for computational purposes seems another issue.

Wikipedia:

Set-theoretical definitions

A function ƒ is an ordered triple of sets (F,X,Y) with restrictions, where

F (the graph) is a set of ordered pairs (x,y),

X (the source) contains all the first elements of F and perhaps more, and

Y (the target) contains all the second elements of F and perhaps more.

The most common restrictions are that F pairs each x with just one y, and that X is just the set of first elements of F and no more.

This definition of function does not contain  "x". Is something like this one what you mean?

"piit" is typeset piit. in Classic

More funny is "pis": colloquial for urine in Spanish...

this help page can be read. I think that it is not available in Mac, so here it is:

theta function,
n. any of a class of special functions that are important in topology, number theory, and analysis. The basic such function from which all others can be derived is the entire function

theta[3](z,q) = 1 + 2*sum((q^(n^2))*cos(2*nz), n=1..infinity)

where q = e^(piit) for im t > 0. When analytic properties are concerned, the dependence on q is suppressed. The special theta function

theta[3](q) = theta[3](0,q) = 1 + 2*sum((q^(n^2)), n=1..infinity)

is the generating function of the sequence of square numbers. It satisfies the remarkable theta transformation formula:

sqrt(s) theta[3][exp(-pis)] = theta[3][exp(-pi/s)].

Compare triple product identity.
 

I guess that you mean

plots[conformal](z^(1/3), z = -1..1);

A nice plot of the cubic roots is got with 'DynamicSystems' (not obvious though):

with( DynamicSystems ):
NewSystem(1/s^3):
RootLocusPlot(%);

were made in recent months in several (sub)threads:

SCRs

Did you report it?

SCR

vote on bugs

As Jacques said:

The discussion happens because of curiosity; filing is 'work'.

Ie, users need "some reward" for reporting. At the minimum "see" that their reports are considered seriously, the bugs are assigned a proper priority and solved in a reasonable time

Now, if exp(x)-13 is an expression and an open term, what does it mean that "there is no denotational semantics" for things like that?

This output seems to be generated in `isolve/inequalities`, as traced by:

infolevel[all]:trace(isolve):
trace(`isolve/inequalities`):
trace(`isolve/isolve`):
isolve({40*a+60*b<=150, a>=0, b>=0});

Yes, it is strange...

about(_NN1);Originally _NN1, renamed _NN1~:
  is assumed to be: AndProp(integer,RealRange(0,infinity))

I have some doubts that kindergarten children are able to solve it.

some things that you say as I am not conversant with CS and its jargon:

while Maple is not at all about functions, it is about expressions.

Isn't Maple also about procedures?

Unfortunately, there is no denotational semantics for open terms! [defn: open term = term with free variables].

Could you give an example of these "open terms"?

I have only a vague idea about lambda calculus. Indeed, what I am reading from the Wikipedia article sounds as if it deals with a (toy?) model for mathematical functions. Presumably, this is what you mean.

So, is there a proof (theorem) showing the imposibility to 'algebraize' calculus?

In case that this was shown, is it equivalent to say that calculus by computer is imposible? (for definiteness,  an ordinary computer, presumably equivalent to a Turing machine).

Yes

plotsetup('ps', 'plotoutput' = "sin3dgrey.ps", 
    'plotoptions' = "color=grey,portrait,noborder"); 
plot3d(sin(x)*sin(y), x = 0..Pi,y=0..Pi,axes=boxed);

show the white axes and tickmarks, only visible against the grey background of the surface

?plot,ps states:

Specifying the keyword "color" enables the plot driver to generate color plots. Using the option "color=value", you can control the color model used to compute and display colors in the output. Accepted values are "rgb", "cmyk", and "gray(or grey)", or "none" for grey scale plots.

I find confusing this last sentence. Sounds to me as if "gray", "grey" and "none" were the same, ie grey scale. But (using Standard GUI):

plotsetup('ps', 'plotoutput' = "sin3dBW.ps", 
    'plotoptions' = "color=none,portrait,noborder"); 
plot3d(sin(x)*sin(y), x = 0..Pi,y=0..Pi);

produces a mesh on black background, while

plotsetup('ps', 'plotoutput' = "sin3dgrey.ps", 
    'plotoptions' = "color=grey,portrait,noborder"); 
plot3d(sin(x)*sin(y), x = 0..Pi,y=0..Pi);

produces a surface image in grey scale.

?odeadvisor,Abel states:

The most general method available at the moment to solve Abel ODEs seems to be the method of "Abel's invariant", described in E. Kamke, p. 26, as sub-method (g) due to M. Chini. ... If the invariant does not depend on x, then the equation can be solved directly.

Clearly, this is a "small" subset of Abel equations.

to see that. For integration of bounded or continuous functions it is OK. But for indefinite integration of diverging functions it does not seem so. So, it depends on what operation (definite or indefinite integration) 'Int' or 'int' represent, the space of functions on which integration is made and the interval in the case of definite integration.  In general, for all the possible integration operations that 'Int' or 'int' represent, it is doubtful that "continuous" makes sense.

I do not expect the Maple property system design to be ever able of dealing with things like the continuity of functionals (maps on function spaces).  May be Axiom?

Ie, the properties that for those 20 names with property 'LinearMap' hold also as true:

`property/ParentTable`[LinearMap];
map(u->`property/ParentTable`[u],%);
map(u->`property/ParentTable`[op(u)],%);
map(u->`property/ParentTable`[op(u)],%);
map(u->`property/ParentTable`[op(u)],%) minus {{}};
map(u->`property/ParentTable`[op(u)],%);
                     {PolynomialMap, StrictlyMonotonic}
                {{monotonic},{InfinitelyDifferentiable}}
                        {{mapping}, {differentiable}}
                          {{TopProp}, {continuous}}
                                 {{mapping}}
                                 {{TopProp}}

There is no representation of domain, hence of its properties (ie nothing like the domain of analytic functions). And the definitions of these properties are given for real functions on the real axis, implicitly defining the domain. So, it seems to me that this property lattice (as here with the family of 'LinearMap') should be corrected to represent the relationship of the properties as they hold on this precise domain. Do you agree?

I wonder also whether the code of Maple somewhere uses these properties implying a different domain.

So, at least,  "analytic" should be added to the property lattice. Probably, all this should be revised.