  We introduce Besov spaces with variable smoothness and integrability by using
the continuous version of Calder\`on reproducing formula. We show that our
space is well-defined, i.e., independent of the choice of basis functions. We
characterize these function spaces by so-called Peetre maximal functions and we
obtain the Sobolev embeddings for these function spaces. We use these results
to prove the atomic decomposition for these spaces.
