  We give a non-perturbative proof of a gradient formula for beta functions of
two-dimensional quantum field theories. The gradient formula has the form
\partial_{i}c = - (g_{ij}+\Delta g_{ij} +b_{ij})\beta^{j} where \beta^{j} are
the beta functions, c and g_{ij} are the Zamolodchikov c-function and metric,
b_{ij} is an antisymmetric tensor introduced by H. Osborn and \Delta g_{ij} is
a certain metric correction. The formula is derived under the assumption of
stress-energy conservation and certain conditions on the infrared behaviour the
most significant of which is the condition that the large distance limit of the
field theory does not exhibit spontaneously broken global conformal symmetry.
Being specialized to non-linear sigma models this formula implies a one-to-one
correspondence between renormalization group fixed points and critical points
of c.
