# Thread: A simple question on multivalued relations

1. ## A simple question on multivalued relations

I have a quick question about single/multi-valued functions.

Given the function $f: \mathbb{R} \to \mathbb{Z}$ where f(x) = the digit in the "tenths" position of x in its decimal representation. An example is $\sqrt{2} \approx 1.4142$ means that $f( \sqrt{2} ) = 4$. It is obvious that this function is single-valued. But I can't find an argument to prove that.

It may well be that I'm over-thinking this.

-Dan

2. ## Re: A simple question on multivalued relations

But what is f(1.0) = f(0.999...): 0 or 9?

3. ## Re: A simple question on multivalued relations

Originally Posted by emakarov
But what is f(1.0) = f(0.999...): 0 or 9?
Nice counter-example! (I'd give you an extra thanks for it, but the Forum won't allow that. So I'll just give you a wet sloppy kiss the next time we have dinner together.)

I can't seem to get my head away from the graph. I'm suspecting a graph is not the way the text wants me to prove multi-valued-ness. Either way, I guess I've got the idea well enough.

(So cool!)

-Dan

4. ## Re: A simple question on multivalued relations

Decimal representation is unique, except for infiniteley repeating nines which take the next highest integer by convention, so that they are also unique.

A sloppy proof of uniqueness is if a has two decimal representations, a-a = 0 is a contradiction.