Summary: In the paper, we construct and study the class of continuous on \([0, 1]\) functions with continuum set of peculiarities (singular, nowhere monotonic, and non-differentiable functionsare among them). The representative of this class is the function \(y = f(x)\) defined by the Engel representation of argument: \[ x = \sum\limits_{n=1}^{\infty} \frac{1}{(2+g_1)(2+g_1 +g_2) \dots (2+g_1 +g_2 +\dots +g_n)}=: \Delta^E_{g_1 g_2 \dots g_n \dots}, \] where \(g_n = g_n (x) \in \{ 0, 1, 2, \dots \}\), and convergent real series \[ \sum\limits_{n=0}^{\infty} u_n = u_0 + u_1 + \dots + u_n + r_n = 1,\qquad \vert u_n \vert < 1, 0 < r_n < 1, \] by the following equality \[ f(\Delta^E_{g_1 (x)g_2 (x)\dots g_n (x)\dots})= r_{g_1 (x)} + \sum\limits_{k=2}^{\infty}\bigg( r_{g_k (x)} \prod\limits_{i=1}^{k-1} u_{g_i (x)} \bigg). \] We study local and global properties of function \(f\): structural, extremal, differential, integral, and fractal properties.