Question

What conditions have to be checked to determine if a function is "well defined"? How would I show that a function is not well defined?

Answer 1

defined on a domain D is said to be a well defined function of the following two conditions are hold :

(i). For all , f(x) is defined .

(ii). For all , f(x) is unique .

If f does not satisfies one of the above two condition is said to be not well defined.

Now I am providing example of two not well defined function

1. Let is fix the domain D = { 1 , 2 , 3 }

{ 1 , 2 , 3} is defined by , f(1) = 2 , f(2) =0 .

This is not a well defined function as it is not defined at 3 that is at 3 f is not defined.

2. { 1 , 2 , 3} is definined by , f(1) = 1 , f(1) = 2 ,f(2) = 3 , f(3)= 0

This is also an not well defined function since f(1) = 1 and 2 that is f(1) is not unique .

N.B - It is important to provide domain to know that whether the function is well defined or not.

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If you have any doubt or need more clarification at any step please let me know in comment box.