Here is the theorem
(a) If $f$ is continuous on an open rectangle $$R : \{ a < x < b , c < y < d \}$$ that contains $(x_0, y_0)$ then the initial value problem $$ y ^ { \prime } = f ( x , y ) , \quad y \left( x _ { 0 } \right) = y _ { 0 } $$ has at least one solution on some open subinterval of $(a, b)$ that contains $x_0$.
(b) If both $f$ and $f_y$ are continuous on $R$ then the equation has a unique solution on some open subinterval of $(a, b)$ that contains $x_0$.
My question is
If the conditions of the Existence and Uniqueness theorem are met, does there exists a unique solution for all $x\in(a,b)$?