  We report the differential charged jet cross section and jet fragmentation
distributions measured with the ALICE detector in proton-proton collisions at a
centre-of-mass energy $\sqrt{s}=$ 7 TeV. Jets with pseudo-rapidity $\left| \eta
\right| < {\rm 0.5}$ are reconstructed from charged particles using the
anti-$k_{\rm T}$ jet finding algorithm with a resolution parameter $R$ = 0.4.
The jet cross section is measured in the transverse momentum interval 5 $\leq
p_{\rm T}^{\rm ch \; jet} <$ 100 GeV/$c$. Jet fragmentation is studied
measuring the scaled transverse momentum spectra of the charged constituents of
jets in four intervals of jet transverse momentum between 5 GeV/$c$ and 30
GeV/$c$. The measurements are compared to calculations from the PYTHIA model as
well as next-to-leading order perturbative QCD calculations with POWHEG +
PYTHIA8. The charged jet cross section is described by POWHEG for the entire
measured range of $p_{\rm T}^{\rm ch \; jet}$. For $p_{\rm T}^{\rm ch \; jet}$
$>$ 40 GeV/$c$, the PYTHIA calculations also agree with the measured charged
jet cross section. PYTHIA6 simulations describe the fragmentation distributions
to 15%. Larger discrepancies are observed for PYTHIA8.
