Thanks for your explanation. I learnt a lot from it. I will do my best to write pure functions.

In many cases, anonymous functions are pure functions, am I right?

I think it is probably true that anonymous functions tend to be pure. Anonymous functions tend to be small. Anonymous functions tend to be created ad hoc for some very specific purpose. These factors make it less likely that you will add side-effects. But I'm not seeing the relevance of the question.

I feel that whether it is a pure function is not important, the more important thing is we make functions to meet our requirements

Okay, we're drifting into philosophical waters. Having worked in the software industry for 25 years, I'm pretty confident in saying that a large proportion of bugs are due to side-effects. Are you sure that "meet our requirements" is straightforward? Let's say you've got a perfectly correct function doing some business computation. Then it's decided that every function must hook into the logging framework in order to get fine-grained observability, so you add a new argument to your function to support logging. Now someone maintaining the logging framework changes some schema expectations, and your function is now broken even though it's still correct per the functional requirements. This kind of thing happens all...the...time.

Look, I'm just pointing out that "pure" and "anonymous" are not equivalent. We can communicate more clearly if we use these terms accurately. I'm also tangentially implying that pure functions are generally less risky in the long run, but your mileage may vary. You're obviously free to use words however you want to, and you're obviously free to write your own code the way you want to. At the end of the day, I'm not invested at all in what decisions you make. Write your functions however you want.

Hello, From your long answer, I learnt a lot. I feel that whether it is a pure function is not important, the more important thing is we make functions to meet our requirements. In many cases, anonymous functions are pure functions, am I right? Thanks.

Simple examples of syntax alternatives that produce the same output:

In[1]:= #^2 &[3]
Out[1]= 9

In[2]:= Function[#^2][3]
Out[2]= 9

In[3]:= Function[x, x^2][3]
Out[3]= 9

Thanks for your help. There are several questions confusing me:

1. I can understand this MySquare3[x_] := (countUsages += 1; x^2) but what is MySquare3 /@ Range[3]. I think iţ should be MySquare3[#] /@ Range[3]. Why can you mitted the [#]

2. What is the difference between

   countUsages = 0;
   MySquare3 /@ Range[3];
   countUsages
   (* 3*)
   

and

    countUsages = 0;
    Function[countUsages += 1; #^2] /@ Range[3];
    countUsages
    (* 3 *)

The result is the same as they all ouput 3.

1. How to tell if there is side effect? Thanks.

I don't know. How to know pure more deep?

I think this thread has conflated pure and anonymous.

Yes. I know now.

Thanks for your reply. But why the Plot3D[Sin[x y], {x, 0, 3}, {y, 0, 3}, ColorFunction -> Function[{x, y, z}, Hue[z]]] doesn't use pure function? Sometimes it uses pure function, but some times it doesn't.