Converse of a well known theorem on integral means

Abstract

0. Introduction. Let f(x) be a real-valued function defined and integrable, in the sense of Lebesgue, in an open interval I of real numbers. If, for h1zO, the points x, x+h lie in I, the integral of f from x to x+lh, divided by h, is called the integral mean of length h of f at x, and will be denoted by AM,hf(x). The integral mean of a function has been a useful tool in various investigations due to the fact that it is smoother than the function itself; for example, if f is integrable in I, its mean will be continuous,2 while if f has a continuous derivative of order n _ 0, the mean will have a continuous derivative of order at least n+1. The thleorem which will be proved is the converse, in a sense, of the above result. We shall show, under a certain restriction (that some restriction is required is evident from simple examples), that the smoothing power of the operation of taking the integral mean is limited; i.e., if f is integrable but not continuous, Mhf will be continuous but not continuously differentiable, while if f has a continuous derivative of order at most n, Mhf will have a continuous derivative of order at most n +1.