# If one wants to intentionally define a multivalued function is it better to use a system of equations or a solution set?

I have heard of a pretty odd concept of having functions which can return multiple values (for instance the square root can return positive or negative values). Is it better to use a system of equations or a "solution set" to describe the function?

For instance, let's say as an example that instead of having the intersection of the two one-sided derivatives (i.e. the regular derivative) we had a function that was the union of the two one-sided derivatives.

This would be either:

As a system of equations:

$$D(f,x) = \lim_{h \to 0^+} \frac {f(x+h) - f(x)}{h} \lor D(f,x) = \lim_{h \to 0^-} \frac {f(x+h) - f(x)}{h}$$

As a solution set:

$$D(f,x) = \{y|\lim_{h \to 0^+} \frac {f(x+h) - f(x)}{h} = y \lor \lim_{h \to 0^-} \frac {f(x+h) - f(x)}{h} = y\}$$

They both seem like valid methods to me. And yes I know functions are supposed to be single-valued but I have heard of multi-valued versions. That is what I am asking about. The derivative is just an example.

• If I ask "What colour is the flag of Japan?" should you say "Either the flag is white or the flag is red", or should you say "The flag has both white and red in it"? – Rahul Apr 24 '17 at 8:07
• @Rahul I fail to see the relevance of that. – The Great Duck Apr 24 '17 at 8:12