  We consider the random iteration of finitely many expanding
$\mathcal{C}^{1+\epsilon}$ diffeomorphisms on the real line without a common
fixed point. We derive the spectral gap property of the associated transition
operator acting on H\"older spaces. As an application we introduce generalised
Takagi functions on the real line and we perform a complete multifractal
analysis of the pointwise H\"older exponents of these functions.
