A sequentially computable function that is not effectively continuous at any point

P. Hertling [Lecture Notes in Computer Science, vol. 2380, Springer, Berlin, 2002, pp. 962–972; Ann. Pure Appl. Logic 132 (2005) 227–246] showed that there exists a sequentially computable function mapping all computable real numbers to computable real numbers that is not effectively continuous. Here, that result is strengthened: a sequentially computable function on the computable real numbers is constructed that is not effectively continuous at any point.