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A problem in FunctionDomain?

Daqing Li
Posted 5 years ago 
FunctionDomain[(x - 5)/Sqrt[x - 5], x]

gives an answer of x >= 5, I think it should be x > 5 instead. Is it a bug?

Thanks.
POSTED BY: Daqing Li
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Hans Milton
Hans Milton
Posted 5 years ago

Going back to Daqings first example

FunctionDomain[(x - 5)/Sqrt[x - 5], x]
(* x >= 5 *)

So 5 is in the domain. But trying it out and actually giving 5 as argument throws errors and returns Indeterminate:

Function[(# - 5)/Sqrt[# - 5]][5]
(* Errors division by zero, complex infinity *)
(* Indeterminate *)

If not a "bug" it is at least a contradiction.
POSTED BY: Hans Milton

It is a very old issue. Try these:

Solve[x^2/x == 0]
Numerator[4/2]
Denominator[6/3]

As functions, x^2/x and x are not the same, but within the algebraic structure of rational functions they are indistinguishable. A "bug" can happen in the frontier cases where Algebra and Analysis disagree. Wolfram perhaps could do more to prevent these embarrassements.
POSTED BY: Gianluca Gorni

I would be careful with the term "bug". I think the result is correct because the limit exists:

Limit[(x - 5)/Sqrt[x - 5], x -> 5]
(* Out:  0   *)
POSTED BY: Henrik Schachner

At some point we are disputing philosophy as to what e.g. x^2/x should mean. But to your (correct) point about general p/q, neighborhoods matter. If these are multivariate, then one has to look further than just the point in question, e.g. (0,0) is not a removable singularity for x/y even though both vanish there.
POSTED BY: Daniel Lichtblau
POSTED BY: Gianluca Gorni
Daqing Li
Daqing Li
Posted 5 years ago

How about the example I posted later? Two equivalent functions just in different input form but the FunctionDomain[] shows two different answers?

POSTED BY: Daqing Li
POSTED BY: Daniel Lichtblau

From the point of view of analysis, x^2/x has a removable singularity.
POSTED BY: Daniel Lichtblau
Daqing Li
Daqing Li
Posted 5 years ago

The two functions should be mathematical equivalent, but the FunctionDomain gives two different domain answer.

POSTED BY: Daqing Li
Daqing Li
Daqing Li
Posted 5 years ago

Or it is more clear as shown here: Screen capture
POSTED BY: Daqing Li
Daqing Li
Daqing Li
Posted 5 years ago

I am convinced that it is a bug as demonstrated from above replies and I tried again:

Screen Copy
POSTED BY: Daqing Li

Even simpler:

FunctionDomain[(2 x - 10)/Sqrt[x - 5], x]
POSTED BY: Gianluca Gorni

The problem is that Mathematica automatically simplifies the fraction

In[6]:= (x - 5)/Sqrt[x - 5]

Out[6]= Sqrt[-5 + x]

before the FunctionDomain does it job. You can bypass this pitfall by giving an equivalent expression that does not trigger the simplification:

FunctionDomain[((x - 5)^2 + Expand[x - 5 - (x - 5)^2])/Sqrt[x - 5], x]
POSTED BY: Gianluca Gorni

Daqing Li, consider

FullSimplify[(x - 5)/Sqrt[x - 5]]

(*Sqrt[-5 + x]*)
POSTED BY: Marvin Ray Burns A.G.S. (cum laude)
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