The authors provide a characterization of convexity for regularly locally Lipschitz functions on Banach spaces by means of a special kind of a second-order upper Dini directional derivative. Let \(X\) be a real Banach space and \(f: X\to\mathbb{R}\) be a locally Lipschitz function. The function \(f\) is assumed to be regular, i.e., the Dini directional derivative \(f'(x; v)\) coincides with the upper Clarke directional derivative \(f^\circ(x; v)\) for all \(x,v\in X\). Introducing the generalized second-order directional derivative \(f^u_{+'}(x; u,v)\) according to \[ f^u_{+'}(x; u,v):= \limsup_{t\downarrow 0}\,{f'(x+ tu; v)- f'(x;v)\over t} \] it is shown in the main theorem of the paper that the function \(f\) is convex if and only if \(f^u_{+'}(x;u,u)\geq 0\) for all \(x,v\in X\). By comparison of this generalized second-order directional derivative with other known generalized second-order directional derivatives, former results regarding the characterization of convex functions can be derived. In the last part of the paper, the authors give some more general characterizations of pseudoconvex and quasiconvex functions.