If any element from the domain associate with more than one element in the range, then it is not a valid function as per the definition. Hence there should not be any f(x) which returns multiple values for any x. The reason is the functions are not associate with a status variable and once a function is defined, the definition remains static.

However your question is closely related generating random numbers. We cannot model a "real" random number generator to work even in a digital computer. What we can really model is just a "pseudo" random generator in which the output sequence will be re-occur after a certain point.

The question is not related to chaos theory as well.

I do not have any idea about your last question.

As Ravimal said, you would lose the mathematical definition of function for such a random behavior. So I am afraid that mathematically speaking this is not possible.

However you can have functions that exhibit a strong chaotic behavior: for a very small variation of the input, you would have a big variation of your output.

The existing algorithms that generate random numbers (e.g. Matlab), generate in reality pseudo-random and pseudo-independent numbers. As an example, you can use the internal clock of a computer and use the date and time as inputs of your algorithm (in addition to the user input) to generate "apparently random" output.

Best,

G.

A true random number generator (not just a pseudo-random number generator) should not exit in real life. If you want a function that never returns the same value no matter how often you evaluate it, that would mean you somehow would have to put in infinite large amount of information into whatever code would describe this function. This is of cause not possible. However, it is also not needed either. Pseudo-random number generators nowadays have such a large period, that the number of times you have to call the function in order for the random numbers to repeat, is so large that for most (or all?) practical purposes they suffice. Furthermore, let us assume in the future you need somehow even more random numbers (because you might need some higher accuracy of some quantity or out of whatever reason), simply writing random number generators with even larger periods should in principle solve the problem. So even if you need more and more random numbers, the finiteness of available random numbers should be no problem for any real application. What might be a greater issue is the speed with which you can generate pseudo-random numbers. I am not an expert in that particular specialisation, but I heard about the idea of using quantum measurements as another ansatz to solve that issue.

So in short: Such a function shouldn't exit. It should not be possible. In principle one could make use of quantum mechanics in order to generate pseudo-random numbers. And the connection of pseudo-random numbers to chaotic dynamical systems has already been raised here:

https://www.researchgate.net/post/On_random_number_generation_via_chaotic_dynamics

In my opinion, Ahmed Fakeih's question simply seems to refer to "function" in the "computer-related" meaning, not mathematical one. By "computer-related" I mean of course a piece of code which returns some value after each execution (functions in programming languages or in spreadsheets). The example of what he wishes to obtain clearly shows (I guess) such an interpretation. The argument "1" in the example is basically meaningless, however,. because basically such a function should run without any parameters,. E.g RAND() function in Excel has no optional or mandatory arguments and just returns a "random" number from the [0,1] and it is claimed that returned numbers have a uniform distribution. However, another Excel function, RANDBETWEEN() has two mandatory arguments "bottom" and "top" which are integers standing, respectively, for the lowest and the largest possible numbers limiting the range from which integer-only numbers are generated. Of course, numbers repeat quite often then. RANDBETWEEN(1,6) emulates tossing a dice and RANDBETWEEN(0,1) emulates tossing a coin.