Approximation in $C^1$ by smooth function while preserving the zero set

Question

My question is in relation to : Approximation by smooth function while preserving the zero set

Let $\mathbb{T}$ denote the unit circle. Given $f \in \mathcal{C}^1(\mathbb{T})$, can we approximate $f$ by smooth functions having the same zero set ? i.e. for $\varepsilon >0$, can we find $g \in \mathcal{C}^\infty(\mathbb{T})$ such that

1. f(x) = 0 if and only if $g(x) = 0$;
2. $\|f-g\|_{\mathcal{C}^1(\mathbb{T})}< \varepsilon$.

Here $\|h\|_{\mathcal{C}^1(\mathbb{T})} := \|h\|_\infty + \|h'\|_\infty = \sup\limits_{z \in \mathbb{T}}|f(z)| + \sup\limits_{z \in \mathbb{T}}|f'(z)|$.