In the paper of Lee et al. an equivalent condition for a function $f$ to be of the first Baire class has been established. This condition is of an $\epsilon-\delta$ type, similarly as in Cauchy's definition of continuity of a function. In the first part of this paper we examine a problem whether it is possible to obtain other classes of functions by further modifications of the above condition. It turns out that, in some sense, the answer is negative. In the second part we consider a topological version of the condition of Lee et al.
"On Cauchy type characterizations of continuity and Baire one functions." Real Anal. Exchange 30 (1) 339 - 346, 2004-2005.