By fractal functions, we mean continuous functions on [0, 1] that have neither finite nor infinite classical derivatives at any point of this interval. Examples are the Weierstrass, van der Waerden, Besicovitch, and other functions. Another example of fractal functions is continuous functions that have no classical derivative at any point of [0, 1] but have a Denjoy–Khinchin approximate derivative almost everywhere (in the Lebesgue measure). An even more “fractal character” is exhibited by functions that are continuous on [0, 1] but have neither finite nor infinite Denjoy–Khinchin approximate derivative at any point x ∈ [0, 1]. In what follows, continuous functions on [0, 1] that have no Denjoy–Khinchin approximate derivative at any point of [0, 1] are called superfractal. Below, we examine some properties of a special class of superfractal functions for which the set of Jarnik points coincides with the interval [0, 1]. It is likely that such functions reflect the limiting boundaries of fractal properties of functions in Banach spaces, such as the lack of derivatives, an unbounded variation in any finite interval [α, β] ⊂ [0, 1], unbounded approximate derivatives of the Dini numbers, and the lack of monotonicity intervals. Despite the “pathological” properties in terms of classical analysis, the above functions can be ones determining finite controls for oscillating systems.