Two classes of Taylor-type formulas for arbitrary continuous functions on intervals are obtained using Bernstein polynomials. These formulas are applicable to both smooth functions and functions that have neither finite nor infinite derivatives at any point. The Taylor-type formulas are considered in close connection with Dini derivatives, which exist for any continuous function. An example is given in which these formulas are applied to the problem of controlling a distributed oscillatory system whose dynamics obeys the d’Alembert representation.