Abstract: 
\def\on#1{\mathop{\rm#1}}Let $C$ denote the Banach space of real-valued
continuous functions on $[0,1]$. Let $\Phi\colon C\times C\to C$. If $\Phi\in
\{ +,\min ,\max\}$ then $\Phi$ is an open mapping but the multiplication $\Phi =\cdot$
is not open.
For an open ball
$B(f,r)$ in $C$ let $B^2(f,r)=B(f,r)\cdot B(f,r)$.
Then
$ f^2\in\on{Int} B^2(f,r)$ for all $r>0$ if and only if
either $f\ge 0$ on $[0,1]$ or $f\le 0$ on $[0,1]$.
Another result states that $\on{Int}(B_1\cdot B_2)\neq\emptyset$ for any two balls $B_1$
and $B_2$ in $C$. We also prove that if
$\Phi\in\{+,\cdot,\min,\max\}$, then
the set $\Phi^{-1}(E)$ is residual whenever $E$ is
residual in $C$.

1. Institute of Mathematics
   Łódź Technical University
   Wólczańska 215
   93-005 Łódź, Poland
   
2. Institute of Mathematics
   Łódź Technical University
   Wólczańska 215
   93-005 Łódź, Poland
   
3. Faculty of Mathematics
   University of Łódź
   Banacha 22
   90-238 Łódź, Poland