Let $\Omega \subset {ℂ}^n$ be a domain with smooth boundary and $p\in \partial \Omega$. A holomorphic function $f$ on $\Omega$ is called a $C^k$ ($k=0,1,2,\cdots$) peak function at $p$ if $f\in C^k\left(\overline{\Omega }\right)$, $f\left(p\right)=1$, and $\left|f\right(q\left)\right|<1$ for all $q\in \overline{\Omega }\setminus \left\{p\right\}$. If $\Omega$ is strongly pseudoconvex, then $C^{\infty }$ peak functions exist. On the other hand, J. E. Fornaess constructed an example in ${ℂ}^2$ to show that this result fails, even for $C^1$ functions, on a weakly pseudoconvex domain [Math. Ann. 227, 173-175 (1977; Zbl 0346.32026)]. Subsequently, E. Bedford and J. E. Fornaess showed that there is always a continuous peak function on a...