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Let $\mathbb{T}$ be the unit circle. Given any closed $A, B\subseteq \mathbb{T}$ with $A \cap B = \emptyset$, there exists a smooth function $\varphi \in \mathcal{C}^\infty(\mathbb{T})$ such that $\varphi = 0$ on $A$ and $\varphi = 1$ on $B$.

Can we have a say on the modulus of the derivative of such a $\varphi$ ? i.e. can with find such a $\varphi$ with $|\varphi'(z)| \leq 1$ for all $z \in \mathbb{T}$ ?

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    Well, such a $\varphi $ would be 1-Lipschitz. Hence, such a function can only exist of $dist (A,B) \geq 1$, and even in this case, it is not clear that such $\varphi $ exists. If the distance is strictly larger then 1, then it does. – 2017-02-07
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    @PhoemueX The question for the case $\operatorname{dist}(A,B)=1$ seems to be somewhat interesting. As far as I can tell, $\phi$ exists iff for all $a\in A, b\in B$ with $d(a,b)=1$, the points are isolated and there is no case of points $b_1,a,b_2$ with $d(b_1,a)=d(a,b_2)=1$ and no $a_1,b,a_2$ with $d(a_1,b)=d(b,a_2)=1$. For in that situation we can find a piecewise linear pre-$\phi$ that has slopes either strictly smaller than $1$ or "overshooting" with slope $1$; this can then be mollified. – 2017-02-07

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