Multiplying balls in the space of continuous functions on [0,1]

Abstract

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 $