On a certain ordering on continuous functions

K.P. Hart, T. Kania and T. Kochanek introduced a certain ordering on a space C(K) (of continuous real-valued functions on a compact space) and investigated spaces K for which C(K) does not have uncountable chains in that ordering. They asked if Eberlein compact spaces had this property. I will answer this question by proving that chains which we consider are always closed and discrete subsets of C(K) with respect to the pointwise-convergence topology.