In this interesting paper the authors study the class \(\mathcal I_c\) of one-component inner functions. This class comprises all inner functions \(\Theta\) whose level set \(\{z\in\mathbb D:|\Theta(z)|<\epsilon\}\) is connected for some \(\epsilon\in ]0,1[\) (hence for all bigger \(\epsilon\)). Their goal is to answer several questions by J. Cima and the rewiever [Complex Anal. Synerg. 3, Paper No. 2, 15 p. (2017; Zbl 1364.30064); in: Advancements in complex analysis. From theory to practice. Cham: Springer. 39–49 (2020; Zbl 1442.30064)]. Using Carleson squares, the authors first give a characterization of the class \(\mathcal I_c\) in terms of the location of the zeros and mass distribution of the singular measures associated with \(\Theta\). Let us give some sample results: if \(B\) is a Blaschke product with infinitely many zeros contained in a Stolz angle with vertex at \(e^{i\theta}\), then \(B\in\mathcal I_c\) if and only if \(\limsup_{r\to 1} |B(re^{i\theta})|<1\). Also, if \(\Theta\) is any inner function whose boundary singular set has Lebesgue measure zero, then there is an interpolating Blaschke product \(B\) such that \(B\Theta\in\mathcal I_c\). It is also shown that singular inner functions associated with a symmetric Cantor measure belong to \(\mathcal I_c\). Finally, if \(\sigma\) is a positive singular measure supported on a closed countable set \(E\subset \mathbb T\) with \(\sigma(\lambda)>0\) for all \(\lambda \in E\), then the associated singular inner function \(I_\sigma\) belongs to \(\mathcal I_c\). An explicit example of a discrete singular inner function is constructed which does not belong to \(\mathcal I_c\).