= Solution
The assertion is understood for every $\eta\in\mathbb R$ and for a topology in which sequentially closed sets are closed, in particular the norm topology of a <Banach space>. Suppose first that $f$ is <sequentially lower semicontinuous> and $x_n\in\Omega_\eta=\{x:f(x)\leq\eta\}$ converges to $x$. Then
$$
f(x)\leq\liminf_nf(x_n)\leq\eta,
$$
so $x\in\Omega_\eta$ and the sublevel set is closed. Conversely, if all sublevel sets are closed but lower semicontinuity fails, there are $x_n\to x$ and a real $\eta<f(x)$ with a subsequence satisfying $f(x_{n_k})\leq\eta$. Closedness of $\Omega_\eta$ would put $x$ in that set, a contradiction. This proves the <closed-sublevel-set characterization of lower semicontinuity>.
Back to article page
