Let $\mathbb{M}$ be the set of real computable numbers, and let $\text{abs, signum} : \mathbb{M} → \mathbb{M}$. They are defined as: $$ \text{abs}(x) = \text{if} \quad x ≥ 0 \quad \text{then} \quad x \quad \text{else} \quad -x $$ $$ \text{signum}(x) = \text{if} \quad x ≠ 0 \quad \text{then} \quad \text{abs}(x)÷x \quad \text{else} \quad 0 $$
Despite that there is no algorithm that compares numbers in $\mathbb{M}$, there is a workaround that proves that $\text{abs}$ is computable:
$$ \text{abs}(x) = \max(x,-x) $$
Since negation and $\max$ are computable, it follows that $\text{abs}$ is computable.
But I don't see a workaround for $\text{signum}$. Does such a workaround exist?
Signum is a discontinuous function, while it can be proved that the set of computable functions is a (proper) subset of the set of continuous function. Hence signum is not computable.