Łojasiewicz inequality

Remarks

Part 1 is the distance inequality and part 2 is the gradient inequality. Łojasiewicz proved them with his theory of semianalytic sets. A later route uses resolution of singularities to make $f$ locally a monomial times a unit, where both inequalities are direct.

Analyticity is necessary. The smooth function with $f(x) = e^{-1/x^2}$ for $x \neq 0$ and $f(0) = 0$ has $Z = \set{0}$. But $\abs{f(x)} / \abs{x}^{\alpha} \to 0$ as $x \to 0$ for every $\alpha > 0$, so part 1 fails on $K = [-1, 1]$.

For $f(x) = x^2$ on $\R$ and $a = 0$, part 2 holds with $\theta = 1/2$ and $C = 1/2$, since $\abs{x^2}^{1/2} = \abs{x} = \tfrac12 \abs{f'(x)}$.

Going deeper

For a student

The gradient inequality is the key to the convergence of gradient flows. If $f$ is real analytic and $x(t)$ is a bounded trajectory of $x' = -\nabla f(x)$, the inequality bounds the length of the trajectory, so $x(t)$ converges to a single critical point. For merely smooth functions a bounded gradient trajectory can accumulate on a whole circle of critical points. The distance inequality is used to divide distributions by real analytic functions.