A borderline case function
$f$
for
${{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$
spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter
$\beta \,>\,0$
. On its support
${{I}_{0}}\,=\,{{\left[ 0,\,1 \right]}^{n}},\,f\left( x \right)$
can be expressed by the binary expansions of the coordinates of
$x$
. In particular,
$f\,=\,{{f}_{\beta }}\,\in \,{{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$
if and only if
$\alpha \,<\,\beta \,<\frac{n}{2}$
, while for
$\beta \,=\,\alpha $
, it was shown by Yue and Dafni that
$f$
satisfies a John–Nirenberg inequality for
${{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$
. When
$\beta \,\ne \,1$
,
$f$
is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside
${{I}_{0}}$
. In addition, it is not monotone along any coordinate direction in any small cube. When the parameter
$\beta \,\in \,\left( 0,\,1 \right)$
,
$f$
is onto from
${{I}_{0}}$
to
$\left[ -\frac{1}{1-{{2}^{-\beta }}},\,\frac{1}{1-{{2}^{-\beta }}} \right]$
, and the graph of
$f$
has a non-integer fractal dimension
$n\,+\,1\,-\beta$
.