Andrei Nikolaevich Schilder (1861-1919) was a Russian landscape painter.
Let \(f:[a,b]\rightarrow\mathbb{R}\) be a real-valued function defined on a bounded closed interval \([a,b]\). We can also consider the limit inferior and limit superior of function \(f\). Now, for \(y\in [a,b]\), we define
\[\limsup_{x\rightarrow y+}f(x)=\inf_{\delta >0}\sup_{0<x-y<\delta}f(x)\mbox{ and }\limsup_{x\rightarrow y-}f(x)=\inf_{\delta >0}\sup_{0<y-x<\delta}f(x)\]
\[\liminf_{x\rightarrow y+}f(x)=\sup_{\delta >0}\inf_{0<x-y<\delta}f(x)\mbox{ and }\liminf_{x\rightarrow y-}f(x)=\sup_{\delta >0}\inf_{0<y-x<\delta}f(x)\]
\[\limsup_{x\rightarrow y}f(x)=\inf_{\delta >0}\sup_{0<|x-y|<\delta}f(x)\mbox{ and }\liminf_{x\rightarrow y}f(x)=\sup_{\delta >0}\inf_{0<|x-y|<\delta}f(x).\]
Let
\[p(\delta)=\sup_{0<|x-y|<\delta}f(x)\mbox{ and }q(\delta)=\inf_{0<|x-y|<\delta}f(x).\]
Then, we see that \(p\) is increasing and \(q\) is decreasing. It follows
\[\limsup_{x\rightarrow y}f(x)=\inf_{\delta >0}\sup_{0<|x-y|<\delta}f(x)= \inf_{\delta >0}p(\delta)=\lim_{\delta\rightarrow 0+}p(\delta)=\lim_{\delta\rightarrow 0+}\sup_{0<|x-y|<\delta}f(x)\]
and
\[\liminf_{x\rightarrow y}f(x)=\sup_{\delta >0}\inf_{0<|x-y|<\delta}f(x)=\sup_{\delta >0}q(\delta)
=\lim_{\delta\rightarrow 0+}q(\delta)=\lim_{\delta\rightarrow 0+}\inf_{0<|x-y|<\delta}f(x).\]
It is easy to see
\[\sup_{0<x-y<\delta}f(x)\leq\sup_{0<|x-y|<\delta}f(x)\mbox{ and }
\sup_{0<y-x<\delta}f(x)\leq\sup_{0<|x-y|<\delta}f(x),\]
which also says
\[\limsup_{x\rightarrow y+}f(x)\leq\limsup_{x\rightarrow y}f(x)\mbox{ and }
\limsup_{x\rightarrow y-}f(x)\leq\limsup_{x\rightarrow y}f(x).\]
It is also easy to see
\[\inf_{0<x-y<\delta}f(x)\geq\inf_{0<|x-y|<\delta}f(x)\mbox{ and }
\inf_{0<y-x<\delta}f(x)\geq\inf_{0<|x-y|<\delta}f(x),\]
which says
\[\liminf_{x\rightarrow y+}f(x)\geq\liminf_{x\rightarrow y}f(x)\mbox{ and }
\liminf_{x\rightarrow y-}f(x)\geq\liminf_{x\rightarrow y}f(x).\]
Proposition 1. Let \(f:[a,b]\rightarrow\mathbb{R}\) be a real-valued function defined on a closed interval \([a,b]\). Then, we have the following results.
(i) \(\limsup_{x\rightarrow y}f(x)\leq A\) if and only if, given \(\epsilon >0\), there exists \(\delta >0\) such that, for all \(x\) with \(0<|x-y|<\delta\), we have \(f(x)\leq A+\epsilon\).
(ii) \(\limsup_{x\rightarrow y}f(x)\geq A\) if and only if, given \(\epsilon >0\) and \(\delta >0\) there exists \(x\) satisfying \(0<|x-y|<\delta\) and \(f(x)\geq A-\epsilon\).
(iii) \(\liminf_{x\rightarrow y}\leq\limsup_{x\rightarrow y}f(x)\) with equality if and only if the limit \(\lim_{x\rightarrow y}f(x)\) exists.
(iv) If \(\limsup_{x\rightarrow y}f(x)=A\) and \({x_{n}}_{n=1}^{\infty}\) is a sequence satisfying \(x_{n}\neq y\) and \(y=\lim_{n\rightarrow\infty}x_{n}\), then \(\limsup_{n\rightarrow\infty}f(x_{n})\leq A\).
(v) If \(\limsup_{x\rightarrow y}f(x)=A\), then there is a sequence \(\{x_{n}\}_{n=1}^{\infty}\) satisfying \(x_{n}\neq y\) with \(y=\lim_{n\rightarrow\infty}x_{n}\) and \(A=\lim_{n\rightarrow\infty}f(x_{n})\).
(vi) For a real number \(l\), we have \(l=\lim_{x\rightarrow y}f(x)\) if and only if \(l=\lim_{n\rightarrow\infty}f(x_{n})\) for every sequence \(\{x_{n}\}_{n=1}^{\infty}\) with \(x_{n}\neq y\) and \(y=\lim_{n\rightarrow\infty}x_{n}\).
Recall the concept of semi-continuity
Proposition 2. Let \(f:[a,b]\rightarrow\mathbb{R}\) be a real-valued function defined on a closed interval \([a,b]\). Then, we have the following results.
(i) The function \(f\) is lower semi-continuous at \(y\) if and only if \(f(y)\leq\liminf_{x\rightarrow y}f(x)\) for \(f(y)\neq -\infty\).
(ii) The function \(f\) is upper semi-continuous at \(y\) if and only if \(f(y)\geq\limsup_{x\rightarrow y}f(x)\) for \(f(y)\neq\infty\).
(iii) The function \(f\) is continuous at \(y\) if and only if it is both lower and upper semi-continuous at \(y\).
(iv) The function \(f\) is lower semi-continuous on the open interval \((a,b)\) if and only if the set \(\{x:f(x)>\lambda\}\) is open for each \(\lambda\in\mathbb{R}\).
(v) The function \(f\) defined on the closed interval \([a,b]\) is lower semi-continuous if and only if there is a monotone increasing sequence \(\{\phi_{n}\}_{n=1}^{\infty}\) of lower semi-continuous step functions on \([a,b]\) satisfying \(f(x)=\lim_{n\rightarrow\infty}\phi_{n}(x)\) for each \(x\in [a,b]\).
