I have two related questions on the representability of integers by quadratic forms in two variables :

(1) Let $f: {\mathbb Z} \times {\mathbb Z} \to {\mathbb Z} $ be such a quadratic form, i.e. we have $f(x,y)=ax^2+bxy+cy^2+dx+ey+g$ for some integer constants $a,b,c,d,e,g$. Suppose that $f$ is not surjective, i.e. some integer is not represented by $f$. Is it true that there is an integer constant $C$ such that in any block of $C$ consecutive integers, at least one of them is not represented by $f$ ?

(2) If the answer to (1) is yes, is there a uniform bound ? In other words, is there a uniform constant $C$ such that for any non-surjective $f$, in any block of $C$ consecutive integers, at least one of them is not represented by $f$ ?

Update : Good answers to my original questions appeared quickly. It seems the only interesting subquestion left is the one asked by fedja : (2') is there a universal $C$ such that for any $f$ with positive definite quadratic part, in any block of $C$ consecutive integers, at least one of them is not represented by $f$ ?

One may also ask, (3) is there a universal $C$ such that for any non-surjective and irreducible $f$, in any block of $C$ consecutive integers, at least one of them is not represented by $f$ ?