Consider the function $f\colon\mathbb{R}\to\{0,1/2,1\}$ defined by $$ f(x):=\begin{cases}1, & x<0\\1/2, & x=0\\0, & x>0\end{cases}. $$ Obviosuly, $M:=\{x\in\mathbb{R}: f(x)=1/2\}=\{0\}$.
Now, for $\varepsilon >0$, I am considering the smooth approximation $$ f^\varepsilon=\eta_\varepsilon * f, $$ where $\eta\colon\mathbb{R}\to\mathbb{R}$ is the mollifier $$ \eta(x):=\begin{cases}C\exp\left(\frac{1}{\lvert x\rvert^2-1}\right), & \lvert x\rvert <1\\ 0, & \lvert x\rvert\geq 1\end{cases} $$ and $\eta_\varepsilon(x):=\frac{1}{\varepsilon}\eta\left(\frac{x}{\varepsilon}\right)$.
I would like to know whether $$ \{x\in\mathbb{R}: f^\varepsilon(x)=1/2\}=M, $$ i.e., whether the position at which the approcimation $f^\varepsilon$ takes the "middle value" 1/2 remains the same position as that for $f$.