Let $U:\mathbb R^2\to\mathbb R$ be a utility function.
If $U$ is strictly increasing and continuous, then it is well known that for any $(x_1,x_2)$ there exists a certainty $(c,c)$ such that $$U(x_1,x_2)=U(c,c).$$
If we assume that $U$ is strictly increasing and upper semi continuous, can we still find a certainty equivalence $(c,c)$ or not?
$U(x_1,x_2)$ is not necessarily expected utility (EU). It could be other utility models under uncertainty, such as the max-min expected utility, choquet expected utility, and others.
For example, see: http://www.columbia.edu/~md3405/BE_Risk_4_17.pdf
Of course one important example of $U$ is the additive representation (subjective expected utility): $U=\sum_ip_iu(x_i)$.
Consider a special case: let $p_1=p_2=0.5$; let $u$ be upper-semi continuous, for example $u(a)=a$ when $a<1$ and $u(a)=a+3$ when $a\geq 1$.
In this case, $U(0,1)=2$.
$U(c,c)=c$ when $c<1$; $U(c,c)=c+3$ when $c\geq 1$. None of the certainty has a utility of 2.
So the answer seems to be "NO"?