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Re: Piecewise vs. /; ?

On 2009.10.15. 13:17, Dan Dubin wrote:
> Folks --
> I would respectfully disagree with the idea that /; is more general
> than Piecewise. The example given, a function that returns a
> different value in the morning than the evening, is simply a
> time-dependent function that could be constructed using Piecewise, as
> in
> f[x_,h_] =  Piecewise[{{x, h<  12}, {-x, h>= 12}}];
> and called via
> f[x,DateList[[4]]]
>    In fact, it seems to me that any difference in the way that
> Mathematica interprets Piecewise and conditionals should be removed
> from later editions of Mathematica, since as far as I can see the
> difference between them is merely a Mathematica-specific  matter of
> the mechanics of how the expressions are evaluated with little (or
> no?) utility, that causes considerable confusion among new users. I
> don't see any benefit in this distinction, but feel free to convince
> me otherwise!

Piecewise[] is explicitly designed to be used to represent a 
mathematical concept, piecewise functions.  Mathematical operations, 
like D[], can be applied to it.

For example,

D[Piecewise[{{1, x > 0}}, 0], x]

/; can be used with patterns (or, as a more special use, with := 
definitions or Module).  It cannot be used to compose a standalone 
mathematical expression like Piecewise can.

Actually Piecewise[] is much closer to Which[] than to /;.  But again, 
Which[] is a programming construct, not something that is meant to be 
used as a mathematical expression.  The following does not work as 
expected from a mathematical point of view:

D[Which[x > 0, 1, True, 0], x]

There are many differences between how these three types of expressions 
are (or are not) evaluated, making Piecewise[] suitable for mathematical 
use, but unwieldy for programming/pattern matching use.

The overlap between programming and mathematical use of some constructs 
(for example, "functions") can already cause confusion in Mathematica. 
I can see how a programming use of Piecewise could be forced... but I 
think that it was a good decision to try to separate these two kinds of 
uses in this case.

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