We show that, up to one exception and as a consequence of first order perturbation theory only, it is impossible that a large portion of the well-known family of breather solutions to the sine Gordon equation could persist under any nontrivial perturbation of the form $$u_{tt} - u_{xx} + \sin u = \varepsilon \Delta \left( u \right) + O\left( {\varepsilon ^2 } \right),$$ where δ is an analytic function in anarbitrarily small neighbourhood ofu=0. Improving known results, we analyze and overcome the particular difficulties that arise when one allows the domain of analyticity of δ to be small. The single exception is a one-dimensional linear space of perturbation functions under which the full family of breathers does persist up to first order in ε.