let $f$ be monotonically increasing on (a,b), if c ∈ (a,b) prove that:
limSup(x) = $f(c^+)$
limInf(x) = $f(c^-)$
I'm not sure where to start, especially that i'm not very profound in fully grasping the Sup and Inf concepts, I sort of always make mistakes. can someone please help me where to start?
Help is greatly appreciated! thank you
If you haven't already covered this, you should start off by showing that $$ f(c^{-}) \leq f(c) \leq f(c^{+})$$ where $f(c^{-}) = \sup\{f(x) : x \in (a, \: c)\}$ and $f(c^{+}) = \inf\{f(x) : x \in (c, \: b)\}$.
Since $f$ is monotone increasing, what can you say about $$\limsup_{x \to c} f(c) = \inf_{\delta > 0} \: (\sup\{f(x) : x \in (c - \delta, c + \delta), x \neq c\})$$