Irregular scaling functions with orthogonal translations

For all integers m,k>1m,k>1 with m≡1modkm≡1modk we construct, among others, a function f:R→R with dense graph in the set [0,k]×R[0,k]×R such thatf(x)=∑n=0m−1f(mx−kn)for everyx∈R,∑i∈Zf(x+i)=1for almost everyx∈R, andf(x)f(x−i)=0for almost everyx∈Rand everyi∈Z∖{0}.