Before revision:  Let \Delta = \{\delta _0,\delta _1,\delta _2,\ldots \} be a countable set of real numbers which has at least m+1 elements. Let f:R^m\rightarrow R be a C^2 function. Let \alpha >0. For each x\in \mathbb{R^m such that \nabla f(x)}  = 0, let \delta (x) = \delta _j, where j is the smallest number so that \nabla ^2f(x)+\delta _j||\nabla f(x)||^{1+\alphaId is invertible.
After revision:  Let \Delta = ^m such that \nabla f(x)} = 0, let \delta (x) = \delta _j, where j is the smallest number so that \nabla ^2f(x)+\delta _j||\nabla f(x)||^{1+\alphaId is invertible.