Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${\left|\nabla f\right|\ge C\left|f\right|}^{\varrho }$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1-R{(n,d)}^{-1}$ with $R(n,d)=d{(3d-3)}^{n-1}$.

We investigate several extension properties of Fréchet differentiable functions defined on closed sets for o-minimal expansions of real closed fields.