For given
p∈[1,∞] and
g∈Lp(R), we establish the existence and uniqueness of solutions
f∈Lp(R), to the equation
where
a∈R,
b∈R∖{0}, and
|a|≠|b|1/p. Solutions include well-known nowhere differentiable functions such as those of Bolzano, Weierstrass, Hardy, and many others. Connections and consequences in the theory of fractal interpolation, approximation theory, and Fourier analysis are established.