Let $\mathbb{D}$ be the open unit disk in the complex plane, and let $\mathrm{Hol}(\mathbb{D})$ denote the class of holomorphic functions in the unit disk. A function $f\in \mathrm{Hol}(\mathbb{D})$ is said to belong to the Bloch space $\mathcal{B}$ if \[ \|f\|_{\mathcal{B}}=|f(0)|+\sup_{z\in\mathbf{D}}(1-|z|^2)|f^\prime (z)|-1$, and $p\in (0, \infty)$, the Dirichlet (type) space $\mathcal{D}^p_\alpha$ consists of all $f\in \mathrm{Hol}(\mathbb D)$ for which \[ \|f\|_{\mathcal{D}^p_\alpha}=|f(0)|+\left (\int_{\mathbb{D}}(1-|z|^2)^\alpha |f^\prime (z)|^pdA(z)\right )^{1/p}1$, a function $f$ belongs to $\mathcal{C}_{\mathcal{B}}(\mathcal{D}^p_{p-1}\cap\mathcal{B})$ if and only if for each $\epsilon >0$, the integral \[ \int_{\Omega_{\epsilon}(f)}\frac{dA(z)}{1-|z|^2} \] is finite, where \[ \Omega_{\epsilon (f)}=\{ z\in \mathbb{D}: (1-|z|^2)|f^{\prime}(z)|\ge \epsilon \}. \] It is also shown that for $p\ge 1$, and $\alpha\in(p-2, p-1]$, $f\in \mathcal{C}_{\mathcal{B}}(\mathcal{D}^p_{\alpha}\cap\mathcal{B})$ if and only if for each $\epsilon >0$, the integral \[ \int_{\Omega_{\epsilon}(f)}\frac{dA(z)}{(1-|z|^2)^{p-\alpha}} \] is finite. In this way, they generalize a recent result proved by \textit{G. Bao} and \textit{N. G. Göğüş} [Complex Anal. Oper. Theory 13, No. 1, 45--59 (2019; Zbl 1417.30053)].