In this note, we show that the assumption of continuity considered in the recent result of Miculescu and Mihail (J Fixed Point Theory Appl, doi:10.1007/s11784-017-0411-7, 2017) can be relaxed further. We also observe that the power convex contraction introduced by Miculescu and Mihail (see condition (1.1)) provides one more solution to the open question of Rhoades (Contemp Math 72:233–245, 1988] regarding existence
of a contractive definition which is strong enough to generate a fixed point but does not force the mapping to be continuous at the fixed point.

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