Let $B_1 : = \{z \in \Bbb C\ |\ |z| \leq 1 \},$ and let $C^0(B_1,\Bbb C)$ be the space of continuous complex valued functions on $B_1$ equipped with the uniform convergence topology. Let $S$ be the set of all smooth functions $f : B_1 \longrightarrow \Bbb C$ such that the normal derivative of $f$ vanishes along the boundary $\partial B_1.$ Is $S$ dense in $C^0 (B_1, \Bbb C)\ $?
If $f \in S$ then what is meant by saying that the normal derivative of $f$ vanishes on the boundary $\partial B_1\ $? I think it means $\frac {\partial\ \mathfrak Rf} {\partial x} = \frac {\partial\ \mathfrak R f} {\partial y} = 0$ and $\frac {\partial\ \mathfrak I f} {\partial x} = \frac {\partial\ \mathfrak I f} {\partial y} = 0$ on $\partial B_1.$ If it is the case then how does it imply denseness of $S$ in $C^0(B_1,\Bbb C)\ $? Any help or suggestion is very much required at this moment.
Thanks for your time.
Source $:$ NBHM PhD Screening Test $2020.$