Exercise 6, § 6

Let $f$ be a real-valued function defined on a topological space $X$. The number (finite or infinite)

is called the oscillation of $f$ at $a \in X$, whenever the right-hand side is defined.

a) Show that $x \to \omega(x; f)$ is upper semi-continuous on the subspace $A$ of $X$ where this mapping is defined.

b) If $f$ is finite on $X$, then $A = X$ and for each $a \in X$ we have

c) Let $f$ be a lower semi-continuous finite real-valued function on $X$. Show that if, at a point $a \in X$, $\omega(a; f)$ has a finite value, then

[Assuming the result false, show that there would exist points $x$ arbitrarily near to $a$ such that $f(x)$ is as large as we please].

d) For each rational number $r = p/q$ in irreducible form (with $q > 0$), put $f(r) = q$. Show that $f$ is lower semi-continuous on $\mathbf{Q}$, but that at each point $r \in \mathbf{Q}$ we have $\omega(r; f) = +\infty$.