Let \(I\) be an interval in \(\mathbb R\). For \(f:I\to\mathbb R\) and \(x\in I\) the pointwise Hölder exponent of \(f\) at \(x\) is defined as \[ h_f(x)=\liminf_{r\to0+}\frac1{\log r}\log \text{Osc}_f(B(x,r)) \] where \[ B(x,r)=[x-r,x+r]\cap I\quad\text{and}\quad \text{Osc}_f(B(x,r))=\sup_{s,t\in B(x,r)} |f(s)-f(t)|. \] Denote by \(E_f(t)\) the level set \(h_f^{-1}(\{t\}),\;t\in\mathbb R^+=[0,\infty)\), and write \[ G_f(t):=\{(x,f(x))\in I\times\mathbb R:x\in E_f(t)\},\quad R_f(t):=\{f(x)\in\mathbb R:x\in E_f(t)\}. \] By analogy with the classical singularity spectrum the author considers the graph and range singularity spectra \(d_f^G\) and \(d_f^R\) defined as \[ d_f^G(t)=\dim_H G_f(t) , \quad d_f^R(t)=\dim_H R_f(t),\quad t\geq0, \] where \(\dim_H\) stands for the Hausdorff dimension. The classical singularity spectrum has an upper bound given by the Legendre transform of the \(L^q\)-spectrum of \(f\). Similarly the author obtains the upper bounds for the graph and range singularity spectra. More precisely, write \[ \tau_f(q)=\liminf_{r\to0+}\frac1{\log r}\log\sup_{\mathcal B}\sum_{B\in\mathcal B}\text{Osc}_f(B)^q,\quad q\in\mathbb R \] where the supremum is taken over all families \(\mathcal B\) of disjoint closed balls \(B\) in \(I\) of radius \(r\) with centers in the set \(\{x\in I:\forall\,r>0,\;\text{Osc}_f(B(x,r))>0\}\). The Legendre transform of \(\tau_f\) is \[ \tau^*_f(t):=\inf_{q\in\mathbb R}(qt-\tau_f(q)). \] It is shown that \[ d_f^G(t)\leq \Big(\frac{\tau^*_f(t)}t\wedge(\tau^*_f(t)+1-t)\Big)\vee\tau_f^*(t)\eqno(1) \] and \[ d_f^R\leq\frac{\tau_f^*(t)}t\wedge1. \eqno(2) \] The main theorem of the paper shows that for the so-called \(b\)-adic independent cascade functions inequalities (1) and (2) turn out to be equalities. The explicit formulation of this result is too awkward for a short review.