  Suppose that $\Omega = \{0, 1\}^ {\mathbb {N}}$ and $ {\sigma}$ is the
one-sided shift. The Birkhoff spectrum $ \displaystyle S_{f}(
{\alpha})=\dim_{H}\Big \{ {\omega}\in {\Omega}:\lim_{N \to \infty} \frac{1}{N}
\sum_{n=1}^N f(\sigma^n \omega) = \alpha \Big \},$ where $\dim_{H}$ is the
Hausdorff dimension. It is well-known that the support of $S_{f}( {\alpha})$ is
a bounded and closed interval $L_f = [\alpha_{f, \min}^*, \alpha_{f, \max}^*]$
and $S_{f}( {\alpha})$ on $L_{f}$ is concave and upper semicontinuous. We are
interested in possible shapes/properties of the spectrum, especially for
generic/typical $f\in C( {\Omega})$ in the sense of Baire category. For a dense
set in $C( {\Omega})$ the spectrum is not continuous on $ {\mathbb {R}}$,
though for the generic $f\in C( {\Omega})$ the spectrum is continuous on $
{\mathbb {R}}$, but has infinite one-sided derivatives at the endpoints of
$L_{f}$. We give an example of a function which has continuous $S_{f}$ on $
{\mathbb {R}}$, but with finite one-sided derivatives at the endpoints of
$L_{f}$. The spectrum of this function can be as close as possible to a
"minimal spectrum". We use that if two functions $f$ and $g$ are close in $C(
{\Omega})$ then $S_{f}$ and $S_{g}$ are close on $L_{f}$ apart from
neighborhoods of the endpoints.
