The authors provide new generalized differentiability notions of first and of second-order for integrable (not necessary continuous) functions \(f:\mathbb{R}^m \to\mathbb{R}\) by means of families of so-called mollifiers and associated sequences of mollified functions. Here, a sequence of mollifiers is a sequence of positive functions \(\{\psi_\varepsilon\}\) with \(\int_{\mathbb{R}^m}\psi_\varepsilon(x)dx=1\) and \(\text{supp}\,\psi_\varepsilon\subset\rho_\varepsilon B\) (where \(B\) is the unit ball of \(\mathbb{R}^m\) and \(\rho_\varepsilon\downarrow 0\) for \(\varepsilon \downarrow 0)\). The associated convolutions \[ f_\varepsilon(x):=\int_{\mathbb{R}^m} f(x-z)\psi_\varepsilon(z)dz \] are called the mollified functions of \(f\). Obviously, the mollified functions \(f_\varepsilon\) are smooth if the mollifiers \(\psi_\varepsilon\) are assumed to be smooth. With respect to such sequences of mollifiers, the (first order) upper and lower mollified directional derivatives of \(f\) according to \[ \overline D_\psi f(x;d):=\sup_{x_n\to x} \limsup_{n\to \infty}\nabla f_{\varepsilon_n}(x_n)^\top d, \] \[ \underline D_\psi f(x; d):=\inf_{x_n\to x}\liminf_{n\to\infty}\nabla f_{\varepsilon_n} (x_n)^\top d, \] the mollified generalized gradient (the mollified subdifferential) of \(f\) according to \[ \partial_\psi f(x):=\left\{\lim_{n\to \infty}\nabla f_{\varepsilon_n}(x_n)\mid x_n\to x\right\} \] and the second-order upper and lower mollified directional derivatives of \(f\) according to \[ \overline D^2_\psi f(x;d,v):=\sup_{x_n\to x}\limsup_{n \to\infty}d^\top\nabla^2f_{\varepsilon_n} (x_n)v, \] \[ \underline D^2_\psi f(x;d,v):=\inf_{x_n\to x}\liminf_{n\to\infty} d^\top \nabla^2f_{\varepsilon_n}(x_n)v, \] are introduced where the last suprema and infima are taken over all possible sequences \(x_n\) converging to \(x\). After discussing the algebraic and topological properties and after comparison with known differentiability concepts, the notions are used for the derivation of necessary and sufficient optimality conditions of first and of second-order for constraint optimization problems and for the second-order characterization of convex functions.