In this paper, we delve into the comprehensive exploration of the continuous curvelet transform (CCT), an advanced iteration of the continuous wavelet transform. Renowned for its applications in diverse mathematical realms such as signal analysis, image processing, and seismic exploration, the CCT holds significant promise. Our focus is on an in-depth examination of the CCT's properties within function spaces, i.e., in Sobolev spaces $H^{s}(\mathbb{R}^{2})$, $W^{m,p}(\mathbb{R}^{2})$, the weighted Sobolev space $W_{\kappa}^{m,p}(\mathbb{R}^{2})$, the generalized Sobolev space $H_{w}^{\omega}(\mathbb{R}^{2})$, Besov space $B^{\alpha, q}_{p}(\mathbb{R}^{2})$, weighted Besov space $B^{\alpha, q}_{p,\kappa}(\mathbb{R}^{2})$, Hardy space $H^{p}(\mathbb{R}^{2})$ and $BMO(\mathbb{R}^{2})$ space. Through investigation, we uncover valuable insights into the continuity and boundedness of the CCT within these function spaces.