Fractal geometry is a valuable formalism for synthesizing and analyzing irregular curves to simulate nonsmooth geometry or roughness. Understanding and controlling these geometries remains challenging because of the complexity of their shapes. This study focuses on the curvature of fractal curves defined from an Iterated Function System (a set of contractive operators). We introduce the Differential Characteristic Function (DCF), a new tool for characterizing and analyzing their differential behavior. We associate a family of DCF to the fixed point of each operator. For each dyadic point of the curve, there exist left and right families of DCF inducing left and right ranges of curvatures: the pseudo-curvatures. A set of illustrations shows the influence of these pseudo-curvatures on the geometry of fractal curves. We propose a first approach for applying our results to roughness generation and control.