Let E ∈ ℝ⁺ be a set consisting of finitely many intervals and a ray [a,∞), and let H _ω ^r be the set of functions defined on E for which |f^r(x) - f^(r) (y)| ≤cfω(|x - y|), where the continuity module ω(x) satisfies the condition ∫^y _oω(x)/x dx + y ∫^∞_yω(x)/x²dx ≤ C0ω(y), y > 0. Let C _σ ^(r,ω) , r > 0, denote the class of entire functions F of order 1/2 and of type σ such that sup|F(z)|̇e-σ|Im √z|z∈C\ℝ (1 + |z|r ω (|z|) + σ -2r ω(σ-2) < <. In the paper, given a function f ∈ H _ω ^r (E), we construct approximating functions F in the class C _σ ^(r,ω) . Approximation by such functions on the set E is analogous to approximation by polynomials on compacts. The analogy involves constructing a scale for measuring approximations and providing a constructive description of the class H _ω ^r (E) in terms of the approximation rate, similar to that of polynomial approximation. Bibliography: 4 titles.