Let Δ(1)(Y ) be a set of all 2𝜋-periodic functions f that are continuous on the real axis R and change their monotonicity at various fixed points yi ∈ [-𝜋; 𝜋); i = 1; :::; 2s; s ∈ N (i.e., there is a set Y := {yi}i∈ℤ of points yi = yi+2s + 2𝜋 on R such that f are nondecreasing on [yi; yi−1] if i is even, and nonincreasing if i is odd). In the article, a function fY = f 2 C(1) ∩ Δ(1)(Y ) has been constructed suchthat$$ \underset{n\to \infty }{\lim}\sup \frac{n{E}_n^{(1)}(f)}{\omega_4\left({f}^{\prime },\pi /n\right)}=\infty, $$where \( {E}_n^{(1)}(f) \) is the error of the best uniform approximation of the function f ∈ Δ(1)(Y) by trigonometric polynomials of order n ∈ N, which also belong to the set Δ(1)(Y ), and 𝜔4(f’; ∙) is the 4-th modulus of smoothness of the function f’. So, for a certain constant c, the inequality \( {E}_n^{(1)}(f)\le \frac{c}{n}{\omega}_3\left({f}^{\prime },\pi /n\right) \) is the best with respect to the order of the modulus of smoothness.