Functions with a Proper Local Maximum in Each Interval

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Let f: R -+ R be a real function. We say that f has a proper local maximum at x e R provided there exists an open neighborhood V of x such that f(y) < f(x) for all y E V (x}. In 1900 A. Schoenflies [3, p. 158] noted that for any functionf, the set of points at whichf has a proper local maximum is at most countable. (This is easy to see: Let B denote the set of all open intervals in R with rational end points; so B is a countable base for the usual topology on R. For each proper local maximum x of f, pick Vx e B such that f(y) < f(x) for all y E Vx (x). The correspondence x -f Vx is one-to-one.) It is trivial to sketch the graph of a continuous function f: R -R which has a proper local maximum at infinitely many points in [0, 1], say at each point in the set (1 /n: n = 1, 2, * } U (0}. This brings up the question: Does there exist a continuous functionf: R -3 R which has a proper local maximum at each point of a countable dense set? It is not easy to visualize the graph of such a continuous function, but the answer to the question is known to be yes. In [1, p. 63, Thm. 3] it was proved that each parameter function of a completely wild arc in R3 will be a continuous function having a proper local maximum at each point of a countable dense set. This result provides an existence theorem for such functions, but the concept of such arcs is not very intuitive. The purpose of this note is to construct, in a straightforward way, a simple example of a continuous function f: R -R which has a proper local maximum at each rational point. This function will be the uniform limit of functions which are easy to visualize. In this sense the construction is analogous to several well-known constructions of continuous nowhere differentiable functions (e.g. [2, p. 141, Thm. 7.18]), and provides a reasonable exercise for a real analysis course.