Q:

There is no graph $G$ with $0<χ_f(G)<1$ or $1<χ_f(G)<2$

There is no graph $G$ with $0<χ_f(G)<1$ or $1<χ_f(G)<2$.
$χ_f$ is the fractional chromatic number

A:

The first one is easy: for any $b$-fold coloring with at least one vertex, we must use at least $b$ colors.  So, if $\chi_f \neq 0$, then $\chi_f \geq b/b = 1$.
The second part is a bit trickier, but still straightforward. Note that if $\chi_f > 1$, then $\chi > 1$, which means that our graph has at least one edge. Show that any $b$-fold coloring of the vertices on that edge uses at least $2b$ colors, from which we conclude that $\chi_f \geq (2b)/b = 2$.

