  A continuous constraint satisfaction problem (CCSP) is a constraint
satisfaction problem (CSP) with an interval domain $U \subset \mathbb{R}$. We
engage in a systematic study to classify CCSPs that are complete of the
Existential Theory of the Reals, i.e., ER-complete. To define this class, we
first consider the problem ETR, which also stands for Existential Theory of the
Reals. In an instance of this problem we are given some sentence of the form
$\exists x_1, \ldots, x_n \in \mathbb{R} : \Phi(x_1, \ldots, x_n)$, where
$\Phi$ is a well-formed quantifier-free formula consisting of the symbols $\{0,
1, +, \cdot, \geq, >, \wedge, \vee, \neg\}$, the goal is to check whether this
sentence is true. Now the class ER is the family of all problems that admit a
polynomial-time many-one reduction to ETR. It is known that NP $\subseteq$ ER
$\subseteq$ PSPACE.
  We restrict our attention on CCSPs with addition constraints ($x + y = z$)
and some other mild technical conditions. Previously, it was shown that
multiplication constraints ($x \cdot y = z$), squaring constraints ($x^2 = y$),
or inversion constraints ($x\cdot y = 1$) are sufficient to establish
ER-completeness. We extend this in the strongest possible sense for equality
constraints as follows. We show that CCSPs (with addition constraints and some
other mild technical conditions) that have any one well-behaved curved equality
constraint ($f(x,y) = 0$) are ER-complete. We further extend our results to
inequality constraints. We show that any well-behaved convexly curved and any
well-behaved concavely curved inequality constraint ($f(x,y) \geq 0$ and
$g(x,y) \geq 0$) imply ER-completeness on the class of such CCSPs.
