Total occupation-time bound for a gradient flow ($T_{\epsilon }\le (\sup f-\inf f)/{\epsilon }^2$)

If $f$ is bounded along a downward gradient flow, the total time spent where $\Vert {\nabla }_{g}f\Vert \ge \epsilon$ is at most $(\sup f-\inf f)/{\epsilon }^2$. The gradient-flow dissipation identity proves this by integrating the indicator of that region. It bounds all visits together, without requiring the visited region to be connected.