$\inf f(A) \leq f( \inf A)$ if $f$ is continuous

Since the domain of $f$ is $[-\infty,\infty]$, something is worth to mention.

For nonempty subset $A$ of $\mathbb{R}$, $\inf A\ne\emptyset$. But it could be the case that $\inf A=-\infty$. So a sequence $(x_{n})\subseteq A$ is such that $x_{n}\rightarrow-\infty$, $f$ being continuous at $-\infty$, we still have $f(x_{n})\rightarrow f(-\infty)$.

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Real Analysis

Supremum And Infimum

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