Let \(f\) be a bounded, real-valued function defined on the real line \(\mathbb{R}\) or \([-\pi,\pi]\). R. N. Mohapatra and R. S. Rodriguez [Math. Nachr. 149, 117-124 (1990; Zbl 0726.42003)] obtained the rate of convergence of singular integrals of \(f\) to \(f\) in the Hölder metric under the assumption: \[ t^{-1} \omega(f;t)\quad\text{is decreasing with }t>0,{(*)} \] where \(\omega(f;t)\) is the modulus of continuity of \(f\).
In the present paper, the author has shown that for \(f\in C_{2\pi}\), the estimates obtained by Mohapatra and Rodriguez (ibid.) may be obtained without using \((*)\).
Remark: The author has mentioned that Mohapatra and Rodriguez (ibid.) has obtained the rate of convergence for \(f\in C_{2\pi}\), which is not correct. In fact, these authors did not use the periodicity of \(f\).