Exercise 2, § 5

a) Let $f$ be a real-valued function defined on a set $X$, let $A$ be a non-empty subset of $X$, and let $\varphi$ be an increasing real-valued function defined on $f(A)$. If $\varphi$ is continuous at the point $a = \sup_{x \in A} f(x)$, then $\sup_{x \in A} \varphi(f(x)) = \varphi(\sup_{x \in A} f(x))$.

b) Let $f$ be a real-valued function defined on a set $X$ filtered by a filter $\mathfrak{F}$, and let $\varphi$ be a real-valued function defined on an open neighbourhood $V$ of the set of cluster points of $f$ with respect to $\mathfrak{F}$. Show that if $\varphi$ is increasing and continuous on $V$, then