The class of everywhere differentiable functions without monotonicity intervals is considered in terms of number theory. A number-theoretic representation of the set of points of the unit interval is constructed using the classification of transcendental numbers proposed by K. Mahler, and a theorem on sufficient conditions for differentiable functions to belong to this class is stated. Results concerning the behavior of derivatives of functions from this class are presented. A mixed problem for the heat equation modeling heat transfer in a distributed system is considered. It is shown that the control function for this system can be everywhere differentiable but having no monotonicity intervals.