# A simple question on multivalued relations

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• Sep 19th 2013, 11:56 AM
topsquark
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
• Sep 19th 2013, 12:55 PM
emakarov
Re: A simple question on multivalued relations
But what is f(1.0) = f(0.999...): 0 or 9?
• Sep 19th 2013, 03:51 PM
topsquark
Re: A simple question on multivalued relations
Quote:

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.) (Heart)

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!) (Nod)

-Dan
• Sep 20th 2013, 07:41 AM
Hartlw
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.