We consider real-valued functions defined on the unit interval. It is known that the class of first-return recoverable functions is the same as the class of polygonally approximable functions and that this class consists of the Baire one functions. Here we introduce the more restrictive classes of consistently first-return recoverable functions and consistently polygonally approximable functions. We show these classes are identical and consist of those functions which are continuous except at countably many points.
"Consistent recovery and polygonal approximation of functions.." Real Anal. Exchange 28 (2) 641 - 648, 2002/2003.