Consider F is an element of C(RxX,Y) such that F(lambda, 0) = 0 for all lambda is an element of R, where X and Y are Banach spaces. Bifurcation from the line Rx{0} of trivial solutions is investigated in cases where F(lambda, center dot ) need not be Frechet differentiable at 0. The main results provide sufficient conditions for mu to be a bifurcation point and yield global information about the connected component of {(lambda,u):F(lambda,u)=0 and u not equal 0}?{(mu,0)} containing (mu, 0). Some necessary conditions for bifurcation are also formulated. The abstract results are used to treat several singular boundary value problems for which Frechet differentiability is not available. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.