Suppose you have a sequence of bounded continuous functions $f_n:(a,b]\to\mathbb{R}$. Then, define the function $$S(x)=\liminf_{n\to\infty}\frac{1}{n}\sum_{j=1}^nf_n(x).$$ Is $S(x)$ continuous?
2026-02-24 16:08:14.1771949294
Continuity of the sequence of averages
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