On a class of fractal functions with graph Hausdorff dimension 2

We prove that the graph of the continuous functiony=Φ(x)=∑k=1∞λ-kαϕλkβx,0⩽x⩽1has Hausdorff dimension 2, where λ > 1, β > α > 1, ϕ(x) = 2x, 0 ⩽ x ⩽ 1/2, ϕ(−x) = ϕ(x) and ϕ(x + 1) = ϕ(x).