Function for which it is unknown whether it is continuous Is there any function $f:\mathbb R\rightarrow \mathbb R$ for which at least some values are known but it is unknown whether $f$ is continuous or not?

Edit: I am looking for examples from actual research, not functions explicitly constructed for that purpose.
Let $f(x)=0$ for $x\neq0$ and $f(0)=$ the first even number that is not the sum of two primes.
Let $f(x)=0$ for $x\neq0$ and $f(0)=$ the first even number that is not the sum of two primes.