Sequential lower semicontinuity

An extended-real functional $f$ is sequentially lower semicontinuous when $x_n\to x$ implies $f(x)\le {\mathrm{liminf}}_{n}f(x_n)$.

A sublevel set consists of points whose objective value is at most a chosen threshold. Coercivity makes finite sublevel sets bounded, while sequential lower semicontinuity makes them sequentially closed sets in the selected topology.

A sublevel set which is compact. A good rate function has this property for every finite level.

On a metric space, a functional is sequentially lower semicontinuous exactly when every sublevel set $\{x:f(x)\le \eta \}$ is closed.