In nonlinear functional analysis course I a studying proper, lower semicontinuous and bounded below functions. I read different criterias for lowersemicontinuous functions, for instance 1) lim inf f(x) at a >=f(a) 2) lim f(x) >= f(a) 3) lim inf f(x)= (min {f(a+), f(a-))>= f(a). lim inf also has many criterion. will someone give me a simpler criterion for lim inf, lim sup, lower semi cont, upper semi cont ?? Thanks in advance
criterion for lower and upper semi continuous functions
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real-analysis
functional-analysis
continuity
1 Answers
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In functional analysis, convex optimization and analysis etc, lower semi-continuity of $f$ at $x_0$ is defined as $$ x_n \to x \ \Rightarrow f(x)\le \lim\inf_{n\to\infty} f(x_n) $$ (if everything is happening in a metric space). See also