A pathological approximately smooth function (Q1073934)

A real-valued measurable function f defined on the real line is said to be smooth if at each point x \[ \lim_{h\to 0}(f(x+h)+f(x-h)-2f(x))/h=0. \] It is known that the set of discontinuities of a smooth function is countable; indeed, this set can be characterized as a scattered set. If one replaces the limit in the above definition by an approximate limit, the notion of approximate smoothness results. Although the set of points of approximate discontinuity of an approximately smooth function must be both of measure zero and of first category, an example is constructed here for which the set of points of approximate discontinuity is uncountable.