An additive $f : \mathbb{R} \to \mathbb{R}$ that is not $x \mapsto cx$: the coefficient of one fixed Hamel basis vector. It is unbounded above and below on every nondegenerate interval, its graph is dense in $\mathbb{R}^{2}$, and every nonempty level set is dense in $\mathbb{R}$

Assume the Axiom of Choice (The Axiom of Choice), which enters through Assuming the Axiom of Choice, $\mathbb{R}$ has a Hamel basis over $\mathbb{Q}$: there is $B \subseteq \mathbb{R}$ such that every real is a finite $\mathbb{Q}$-linear combination of elements of $B$ in exactly one way, and each basis vector carries a well-defined $\mathbb{Q}$-linear coefficient map and hence through Zorn's lemma. Fix a Hamel basis $B$ of $\mathbb{R}$ over the canonical copy $\mathbb{Q} \subseteq \mathbb{R}$ of the rationals (The rationals embed densely in the reals, A field is a vector space over itself, and over any subfield $K \subseteq F$ every $F$-vector space is a $K$-vector space by restricting the scalars, Vector space over a field), fix $b_{\star} \in B$, and let

$f \;:=\; \Lambda_{b_{\star}} : \mathbb{R} \to \mathbb{R}$

be the coefficient map of $b_{\star}$ (Assuming the Axiom of Choice, $\mathbb{R}$ has a Hamel basis over $\mathbb{Q}$: there is $B \subseteq \mathbb{R}$ such that every real is a finite $\mathbb{Q}$-linear combination of elements of $B$ in exactly one way, and each basis vector carries a well-defined $\mathbb{Q}$-linear coefficient map, claim 4). Write $W := W_{b_{\star}} = \operatorname{span}(B \setminus \{b_{\star}\})$ (Linear combination of a finite list, and the span $\operatorname{span}(S)$ as the smallest linear subspace containing $S$). Then:

1. $f$ is additive (Cauchy's functional equation $f(x+y) = f(x) + f(y)$, and the additive functions $\mathbb{R} \to \mathbb{R}$) and is not of the form $x \mapsto cx$ for any real $c$ (FALSE: every additive $f : \mathbb{R} \to \mathbb{R}$ is of the form $x \mapsto cx$ for a single real $c$);
2. $f$ is bounded neither above nor below on any nondegenerate interval (Lower bound, bounded below, bounded set, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), is monotone on no nondegenerate interval (Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$, with the dictionary to monotone sequences), is of constant sign on none, and is continuous at no point of $\mathbb{R}$ (Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point);
3. the graph $\{(x,f(x)) : x \in \mathbb{R}\}$ is dense in $\mathbb{R}^{2}$ for the metric $d_\infty$ ($\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it, Interior, closure, boundary, limit point, isolated point and dense subset of a metric space);
4. the values of $f$ are exactly the rationals, and for every rational $r$ the level set $f^{-1}(\{r\}) = \{\, x \in \mathbb{R} : f(x) = r \,\}$ is dense in $\mathbb{R}$; for an irrational $v$ the level set $f^{-1}(\{v\})$ is empty.

Claim 2 is the contrapositive of Six regularity conditions each force an additive $f : \mathbb{R} \to \mathbb{R}$ to be $x \mapsto f(1)x$: continuity at a single point, monotonicity on a nondegenerate interval, boundedness above on one, boundedness below on one, constancy of sign on one, and a graph that is not dense in $\mathbb{R}^{2}$ applied to claim 1, clause by clause, and claim 3 is the contrapositive of its sixth clause.

• The pathology is entirely a consequence of the two facts in claim 1. The proof uses nothing about $f$ except that it is additive and not linear; every other property is read off Six regularity conditions each force an additive $f : \mathbb{R} \to \mathbb{R}$ to be $x \mapsto f(1)x$: continuity at a single point, monotonicity on a nondegenerate interval, boundedness above on one, boundedness below on one, constancy of sign on one, and a graph that is not dense in $\mathbb{R}^{2}$ by contraposition. A single such function therefore witnesses the failure of all six regularity conditions at once.

• What the level sets look like. They are the cosets of the $\mathbb{Q}$-subspace $W = \ker f$, one for each rational value, and each is dense. So $\mathbb{R}$ is partitioned into countably many dense sets, on each of which $f$ is constant. The companion function of A bounded function on $\mathbb{R}$ with no local maximum and no local minimum at any point, upper semicontinuous at no point and lower semicontinuous at no point: compose the Hamel coefficient with a strictly increasing injection of $\mathbb{R}$ into $(0,1)$ is built by relabelling those values.

• No measurability claim is made. The classical statement that a Hamel coefficient map is not Lebesgue measurable is not asserted here: this library develops no measure as it stands, so the statement is not expressible, and nothing above depends on it.