For Bernstein operators \(B_ n\) one has the direct theorem \[ \| B_ n f-f\|_ \infty\leq C\left\{ w^ 2_ \varphi\left(f,{1\over\sqrt n}\right)_ \infty +{1\over n} \| f\|_ \infty\right\},\quad n\in\mathbb{N}, \] where \(C\) is independent of \(n\) and \(w^ 2_ \varphi(f,\cdot)_ \infty\) denotes the second order Ditzian-Totik modulus of smoothness with respect to the sup-norm and here with the special stepweight \(\varphi(x)=\sqrt{x(1-x)}\). Neither for Bernstein- Kantorovich operators \(K_ n\) nor for Bernstein-Durrmeyer operators \(M_ n\) on \(C[0,1]\) an estimate with the same upper bound does hold. The latter shows a counterexample in the present paper. Writing \(L_ n\) for either \(K_ n\) or \(M_ n\) the authors prove the following direct theorem on \(C[0,1]\): \[ \| L_ n f-f\|_ \infty\leq C\left\{w^ 2_ \varphi\left(f,{1\over\sqrt n}\right)_ \infty+w\left(f,{1\over n}\right)_ \infty+{1\over n}\| f\|_ \infty\right\},\quad n\in \mathbb{N}, \] where \(C\) is again independent of \(n\) and \(w(f,\cdot)_ \infty\) denotes the classical first order modulus of continuity, \(\varphi(x)=\sqrt{x(x-1)}\). Reviewer's remark'' For \(L_ n=M_ n\) and simultaneous approximation \textit{H. H. Gonska} and \textit{X. L. Zhou} have obtained already in 1990 the same upper bound [c.f. J. Approximation Theory 67, No. 3, 284-302 (1991; Zbl 0756.41027)].