Dimensions of graphs of prevalent continuous maps

Let K be an uncountable compact metric space and let (K,R-d) denote the set of continuous maps f : K -> R-d endowed with the maximum norm. The goal of this paper is to determine various fractal dimensions of the graph of a prevalent f is an element of C (K,R-d). As the main result of the paper we show that if K has at most finitely many isolated points then the lower and upper box dimension of the graph of a prevalent f is an element of C (K,R-d) are (dim) under bar K-B + d and (dim) under bar K-B + d, respectively. This generalizes a theorem of Gruslys, Jonusas, Mijovic, Ng, Olsen, and Petrykiewicz. We prove that the packing dimension of the graph of a prevalent f is an element of C (K,R-d) is dim(P) K + d, generalizing a result of Balka, Darji, and Elekes. Balka, Darji, and Elekes proved that the Hausdorff dimension of the graph of a prevalent f is an element of C (K,R-d) equals dim(H) K + d. We give a simpler proof for this statement based on a method of Fraser and Hyde.