02/19/2008, 01:20 PM
So my continuous counter proposal for zeration would be the following operation:
\( a\circ b = \begin{cases}b+1 & : b>a+1\\a+2&: b\le a+1\end{cases} \)
This is continuous in both arguments and satisfies
\( a\circ a=a+2 \)
\( a\circ (a\circ a)=a+3 \)
\( a\circ (a\circ (a\circ a))=a+4 \)
etc. for all real \( a \)
\( a\circ b = \begin{cases}b+1 & : b>a+1\\a+2&: b\le a+1\end{cases} \)
This is continuous in both arguments and satisfies
\( a\circ a=a+2 \)
\( a\circ (a\circ a)=a+3 \)
\( a\circ (a\circ (a\circ a))=a+4 \)
etc. for all real \( a \)